Chapter 24 - Stochastic Processes.ipynb

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   "source": [
    "## 24. Stochastic Processes"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "### 24.1 Introduction"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "A **stochastic process** $\\{ X_t : t \\in T \\}$  is a collection of random variables.  We shall sometimes write $X(t)$ instead of $X_t$.  The variables $X_t$ take values in some set $\\mathcal{X}$ called the **state space**.  The set $T$ is called the **index set** and for our purposes can be thought of as time.  The index set can be discrete, $T = \\{0, 1, 2, \\dots \\}$ or continuous $T = [0, \\infty)$ depending on the application."
   ]
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   "source": [
    "Recall that if $X_1, \\dots, X_n$ are random variables then we can write the joint density as\n",
    "\n",
    "$$ f(x_1, \\dots, x_n) = f(x_1) f(x_2 | x_1) \\dots f(x_n | x_1, \\dots, x_{n-1}) = \\prod_{i=1}^n f(x_i | \\text{past}_i) $$\n",
    "\n",
    "where $\\text{past}_i$ refers to all variables before $X_i$."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 24.2 Markov Chains"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The process $\\{ X_n : n \\in T \\}$ is a **Markov Chain** if\n",
    "\n",
    "$$ \\mathbb{P}(X_n = x | X_0, \\dots, X_{n-1}) = \\mathbb{P}(X_n = x | X_{n-1})$$\n",
    "\n",
    "for all $n$ and for all $x \\in \\mathcal{X}$."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a Markov chain, the joint density function can be written as\n",
    "\n",
    "$$ f(x_1, \\dots, x_n) = f(x_1) f(x_2 | x_1) f(x_3 | x_2) \\dots f(x_n | x_{n - 1}) $$\n",
    "\n",
    "A Markov chain can be represented by the following DAG:\n",
    "\n",
    "$$ X_1 \\longrightarrow X_2 \\longrightarrow X_3 \\longrightarrow \\cdots \\longrightarrow X_n \\longrightarrow \\cdots $$\n",
    "\n",
    "Each variable has a single parent, namely, the previous observation."
   ]
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   "cell_type": "markdown",
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   "source": [
    "The theory of Markov chains is very rich and complex.  Our goal is to answer the following questions:\n",
    "\n",
    "1. When does a Markov chain \"settle down\" into some sort of equilibrium?\n",
    "\n",
    "2. How do we estimate the parameters of a Markov chain?\n",
    "\n",
    "3. How can we construct Markov chains that converge to a given equilibrium and why would we want to do that?\n",
    "\n",
    "Questions 1 and 2 will be approached this chapter, question 3 in the next chapter."
   ]
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  {
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   "execution_count": 1,
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   "source": [
    "import numpy as np\n",
    "\n",
    "def generate_random_walk(n, seed=None):\n",
    "    if seed is not None:\n",
    "        np.random.seed(seed)\n",
    "    \n",
    "    X = np.empty(n)\n",
    "    X[0] = 0\n",
    "    for i in range(1, n):\n",
    "        X[i] = X[i - 1] + np.random.uniform(low=-1, high=1)\n",
    "    \n",
    "    return X\n",
    "\n",
    "def generate_random_walk_bound(n, drift=-0.4, min_value=-10, seed=None):\n",
    "    if seed is not None:\n",
    "        np.random.seed(seed)\n",
    "    \n",
    "    X = np.empty(n)\n",
    "    X[0] = 0\n",
    "    for i in range(1, n):\n",
    "        X[i] = max(X[i - 1] + drift, min_value) + np.random.uniform(low=-1, high=1)\n",
    "    \n",
    "    return X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 864x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "\n",
    "A = generate_random_walk(1000, seed=0)\n",
    "B = generate_random_walk_bound(1000, seed=0)\n",
    "\n",
    "ax = plt.subplot(2, 1, 1)\n",
    "ax.plot(np.arange(0, len(A)), A)\n",
    "ax.set_title('Does not settle into equilibrium')\n",
    "\n",
    "ax = plt.subplot(2, 1, 2)\n",
    "ax.plot(np.arange(0, len(B)), B)\n",
    "ax.set_title('Does settle into equilibrium')\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Transition Probabilities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The key quantities of a Markov chain are the probabilities of jumping from one state into another state.\n",
    "\n",
    "We call \n",
    "\n",
    "$$ \\mathbb{P}(X_{n+1} = j | X_n = i) $$\n",
    "\n",
    "the **transition probabilities**.  If the transition probabilities do not change with time, we say the chain is **homogeneous**.  In this case we define $p_{ij} = \\mathbb{P}(X_{n+1} = j | X_n = i)$.  The matrix $P$ whose $(i, j)$ element is $p_{ij}$ is called the **transition matrix**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will only consider homogeneous chains.  Notice how each $P$ has two properties: (i) $p_{ij} \\geq 0$ and (ii) $\\sum_i p_{ij} = 1$.  Each row is a probability mass function.  A matrix with these properties is called a **stochastic matrix**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let\n",
    "\n",
    "$$ p_{ij}(n) = \\mathbb{P}(X_{m + n} = j | X_{m} = i) $$\n",
    "\n",
    "be the probability of going from state $i$ to state $j$ in $n$ steps.  Let $P_n$ be the matrix whose $(i, j)$ element is $p_{ij}(n)$.  These are called the **$n$-step transition probabilities**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 24.9 (The Chapman-Kolmogorov equations)**.  The $n$-step probabilities satisfy\n",
    "\n",
    "$$ p_{ij}(m + n) = \\sum_k p_{ij}(m) p_{kj}(n) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proof**.  Recall that, in general,\n",
    "\n",
    "$$ \\mathbb{P}(X = x, Y = y) = \\mathbb{P}(X = x) \\mathbb{P}(Y = y | X = x) $$\n",
    "\n",
    "This fact is true when conditioned in another variable,\n",
    "\n",
    "$$ \\mathbb{P}(X = x, Y = y | Z = z) = \\mathbb{P}(X = x | Z = z) \\mathbb{P}(Y = y | X = x, Z = z) $$\n",
    "\n",
    "Also, recall the law of total probability:\n",
    "\n",
    "$$ \\mathbb{P}(X = x) = \\sum_y \\mathbb{P}(X = x, Y = y) $$\n",
    "\n",
    "Using these facts and the Markov property we have:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "p_{ij}(m + n) &= \\mathbb{P}(X_{m + n} = j | X_0 = i) \\\\\n",
    "&= \\sum_k \\mathbb{P}(X_{m + n} = j, X_m = k | X_0 = i) \\\\\n",
    "&= \\sum_k \\mathbb{P}(X_{m + n} = j | X_m = k, X_0 = i) \\mathbb{P}(X_m = k | X_0 = i) \\\\\n",
    "&= \\sum_k \\mathbb{P}(X_{m + n} = j | X_m = k) \\mathbb{P}(X_m = k | X_0 = i) \\\\\n",
    "&= \\sum_k p_{ik}(m) p_{kj}(n)\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this definition is equivalent to matrix multiplication; hence we have shown that\n",
    "\n",
    "$$ P_{m + n} = P_m P_n $$\n",
    "\n",
    "By definition, $P_1 = P$.  Using the above theorem, we get\n",
    "\n",
    "$$ P_n = P^n \\equiv \\underbrace{P \\times P \\times \\cdots \\times P}_{\\text{multiply matrix } n \\text{ times}} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $\\mu_n = (\\mu_n(1), \\dots, \\mu_n(N))$ be a row vector where\n",
    "\n",
    "$$ \\mu_n(i) = \\mathbb{P}(X_n = i) $$\n",
    "\n",
    "is the marginal probability that the chain is in state $i$ at time $n$.  In particular, $\\mu_0$ is called the **initial distribution**.  To simulate a Markov chain, all you need to know is $\\mu_0$ and $P$.  The simulation would look like this:\n",
    "\n",
    "1. Draw $X_0 \\sim \\mu_0$.  Thus, $\\mathbb{P}(X_0 = i) = \\mu_0(i)$.\n",
    "2. Suppose the outcome of step 1 is $i$.  Draw $X_1 \\sim P$. In other words, $\\mathbb{P}(X_1 = j | X_0 = i) = p_{ij}$.\n",
    "3. Suppose the outcome of step 2 is $j$.  Draw $X_2 \\sim P$. In other words, $\\mathbb{P}(X_2 = k | X_1 = j) = p_{jk}$.\n",
    "\n",
    "and so on."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It might be difficult to understand the meaning of $\\mu_n$.  Imagine simulating the chain many times.  Collect all of the outcomes at time $n$ from all the chains.  This histogram would look approximately like $\\mu_n$.  A consequence of the previous theorem is the following:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Lemma 24.10**.  The marginal probabilities are given by\n",
    "\n",
    "$$ \\mu_n = \\mu_0 P^n $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proof**.\n",
    "\n",
    "$$ \\mu_n(j) = \\mathbb{P}(X_n = j) = \\sum_i \\mathbb{P}(X_n = j | X_0 = i) \\mathbb{P}(X_0 = i) = \\sum_i \\mu_0(i) p_{ij}(n) = \\mu_0 P^n $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Summary**\n",
    "\n",
    "1.  Transition matrix:  $P(i, j) = \\mathbb{P}(X_{n+1} = j | X_n = i)$\n",
    "2.  $n$-step matrix: $P_n(i, j) = \\mathbb{P}(X_{m+n} = j | X_m = i)$\n",
    "3.  $P_n = P^n$\n",
    "4.  Marginal probabilities:  $\\mu_n(i) = \\mathbb{P}(X_n = i)$\n",
    "5.  $\\mu_n = \\mu_0 P^n$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### States"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The states of a Markov chain can be classified according to various properties.\n",
    "\n",
    "We say that $i$ **reaches** $j$ (or $j$ is **accessible** from $i$) if $p_{ij}(n) > 0$ for some $n$, and we write $i \\rightarrow j$.  If $i \\rightarrow j$ and $j \\rightarrow i$ then we write $i \\leftrightarrow j$ and we say that $i$ and $j$ **communicate**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 24.12**.  The communication relation satisfies the following properties:\n",
    "\n",
    "1.  $i \\leftrightarrow i$.\n",
    "2.  If $i \\leftrightarrow j$ then $j \\leftrightarrow i$.\n",
    "3.  If $i \\leftrightarrow j$ and $j \\leftrightarrow k$ then $i \\leftrightarrow k$.\n",
    "4.  The set of states $\\mathcal{X}$ can be written as a disjoint union of **classes** $\\mathcal{X} = \\mathcal{X}_1 \\cup \\mathcal{X}_2 \\cup \\cdots$  where two states $i$ and $j$ communicate with each other if and only if they are in the same class."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If all states communicate with each other then the chain is called **irreducible**.  A set of states is **closed** if once you enter that set states you never leave.  A closet set consisting of a single state is called an **absorbing state**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we start a chain in state $i$.  Will the chain ever return to state $i$?  If so, that state is called persistent or recurrent.\n",
    "\n",
    "State $i$ is **recurrent** or **persistent** if \n",
    "\n",
    "$$ \\mathbb{P}(X_n = i \\text{ for some } n \\geq 1 | X_0 = i) = 1 $$\n",
    "\n",
    "Otherwise, state $i$ is **transient**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 24.15**.  A state $i$ is recurrent if and only if\n",
    "\n",
    "$$ \\sum_n p_{ii}(n) = \\infty $$\n",
    "\n",
    "A state $i$ is transient if and only if\n",
    "\n",
    "$$ \\sum_n p_{ii}(n) < \\infty $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proof**.  Define\n",
    "\n",
    "$$ \n",
    "I_n = \\begin{cases}\n",
    "1 & \\text{if } X_n = i \\\\\n",
    "0 & \\text{if } X_n \\neq i\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "The number of times that the chain is in state $i$ is $Y = \\sum_{n=0}^\\infty I_n$.  The mean of $Y$, given that the chain starts in state $i$, is\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbb{E}(Y | X_0 = i) &= \\sum_{n=0}^\\infty \\mathbb{E}(I_n | X_0 = i) \\\\\n",
    "&= \\sum_{n=0}^\\infty \\mathbb{P}(X_n = i | X_0 = i) \\\\\n",
    "&= \\sum_{n=0}^\\infty p_{ii}(n)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Define $a_i = \\mathbb{P}(X_n = i \\text{ for some } n \\geq  1 | X_0 = i)$.  If $i$ is recurrent, $a_i = 1$.  Thus, the chain will eventually return to $i$.  Once it does, we argue again that since $a_i = 1$, the chain will return to state $i$ again.  By repeating this argument, we conclude that $\\mathbb{E}(Y | X_0 = i) = \\infty$.\n",
    "\n",
    "If $i$ is transient, then $a_i < 1$.  When the chain is in state $i$, there is a probability $1 - a_i > 0$ that it will never return to state $i$.  Thus, the probability that the chain is in state $i$ exactly $n$ times is $a_i^{n - 1}(1 - a_i)$.  This is a geometric distribution that has finite mean."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 24.16**.  Facts about recurrence:\n",
    "\n",
    "1.  If a state $i$ is recurrent and $i \\leftrightarrow j$ then $j$ is recurrent.\n",
    "2.  If a state $i$ is transient and $i \\leftrightarrow j$ then $j$ is transient.\n",
    "3.  A finite Markov chain must have at least one recurrent state.\n",
    "4.  The states of a finite, irreducible Markov chain are all recurrent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 24.17 (Decomposition Theorem)**.  The state space $\\mathcal{X}$ can be written as the disjoint union\n",
    "\n",
    "$$ \\mathcal{X} = \\mathcal{X}_{T} \\cup \\mathcal{X}_{1} \\cup \\mathcal{X}_{2} \\cup \\cdots $$\n",
    "\n",
    "where the $\\mathcal{X}_T$ are the transient states and each $\\mathcal{X}_i$ is a closed, irreducible set of recurrent states."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Convergence of Markov Chains"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose that $X_0 = i$.  Define the **recurrence time**\n",
    "\n",
    "$$ T_{ij} = \\min \\{ n > 0 : X_n = j \\} $$\n",
    "\n",
    "assuming $X_n$ ever returns to the state $i$, otherwise define $T_{ij} = \\infty$.  The **mean recurrence time** of a recurrent state $i$ is\n",
    "\n",
    "$$ m_i = \\mathbb{E}(T_{ii}) = \\sum_n n f_{ii}(n) $$\n",
    "\n",
    "where\n",
    "\n",
    "$$ f_{ij}(n) = \\mathbb{P}(X_1 \\neq j, X_2 \\neq j, \\dots, X_{n-1} \\neq j, X_n \\neq j | X_0 = i) $$\n",
    "\n",
    "A recurrent state is **null** if $m_i = \\infty$, otherwise it is called **non-null** or **positive**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Lemma 24.18**.  If a state is null and recurrent, then $p_{ii}^n \\rightarrow 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Lemma 24.19**.  In a finite state Markov chain, all recurrent states are positive."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a three state chain with transition matrix\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "0 & 1 & 0 \\\\\n",
    "0 & 0 & 1 \\\\\n",
    "1 & 0 & 0\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Suppose we start the chain in state 1.  Then we will be in state 3 at times $3, 6, 9, \\dots$.  This is an example of a periodic chain.  Formally, the **period** of state $i$ is $d$ is $p_{ii}(n) = 0$ whenever $n$ is not divisible by $d$ and $d$ is the largest integer with this property.  Thus, $d = \\text{gcd} \\{ n : p_{ii}(n) = 0 \\}$, where gcd means \"greatest common divisor\".  State $i$ is **periodic** if $d(i) > 1$ and **aperiodic** if $d(i) = 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Lemma 24.20**.  If a state $i$ has period $d$ and $i \\leftrightarrow j$ then $j$ has period $d$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A state is **ergodic** if it is recurrent, non-null and aperiodic.  A chain is ergodic if all its states are ergodic."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $\\pi = (\\pi_i : i \\in \\mathcal{X})$ be a vector of non-negative numbers that sum to one.  Thus $\\pi$ can be thought of as a probability mass function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We say that $\\pi$ is a **stationary** (or **invariant**) distribution if $\\pi = \\pi P$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is the intuition.  Draw $X_0$ from distribution $\\pi$ and suppose that $\\pi$ is a stationary distribution.  Now draw $X_1$ according to the transition probability of the chain.  The distribution of $X_1$ is then $\\mu_1 = \\mu_0 P = \\pi P = \\pi$.  Continuing this way, the distribution of $X_n$ is $\\mu_n = \\mu_0 P^n = \\pi P^n = \\pi$.  In other words:  if at any time the chain has distribution $\\pi$, then it will continue to have distribution $\\pi$ forever."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We say that a chain has **limiting distribution** if\n",
    "\n",
    "$$ P^n \\rightarrow \\begin{bmatrix}\n",
    "\\pi \\\\ \\pi \\\\ \\vdots \\\\ \\pi\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "for some $\\pi$.  In other words, $\\pi_j = \\lim_{n \\rightarrow \\infty} P_{ij}^n$ exists and is independent of $i$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 24.24**.  An irreducible, ergodic Markov chain has a unique stationary distribution $\\pi$.  The limiting distribution exists and is equal to $\\pi$.  If $g$ is any bounded function, then, with probabiity 1,\n",
    "\n",
    "$$ \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\sum_{n=1}^N g(X_n) = \\mathbb{E}_\\pi(g) \\equiv \\sum_j g(j) \\pi_j $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The last statement of the theorem is the law of large numbers for Markov chains.  It says that sample averages converge to their expectations.  Finally, there is a special condition which will be useful later.  We say that $\\pi$ satisfies **detailed balance** if\n",
    "\n",
    "$$ \\pi_i p_{ij} = p_{ji} \\pi_j $$\n",
    "\n",
    "Detailed balance guarantees that $\\pi$ is a stationary distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 24.25**.  If $\\pi$ satisfies detailed balance then $\\pi$ is a stationary distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proof**.  We need to show that $\\pi P = \\pi$.  The $j$-th element of $\\pi P$ is $\\sum_i \\pi_i p_{ij} = \\sum_i p_{ji}\\pi_j  = \\pi_j \\sum_i p_{ji} = \\pi_j$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The importance of detailed balance will become clear when we discuss Markov chain Monte Carlo methods.\n",
    "\n",
    "**Warning**: Just because a chain has a stationary distribution does not mean it converges."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Inference for Markov Chains"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a chain with finite state space $\\mathcal{X} = \\{ 1, 2, \\dots, N \\} $.  Suppose we observe $n$ observations $X_1, \\dots, X_n$ from this chain.  The unknown parameters of a Markov chain are the initial probabilities $\\mu_0 = (\\mu_0(1), \\mu_0(2), \\dots)$ and the elements of the transition matrix $P$.  Each row of $P$ is a multinomial distribution, so we are essentially estimating $N$ distributions (plus the initial probabilities).  Let $n_{ij}$ be the observed number of transitions from state $i$ to state $j$.  The likelihood function is\n",
    "\n",
    "$$ \\mathcal{L}(\\mu_0, P) = \\mu_0(x_0) \\prod_{r=1}^n p_{X_{r - 1}, X_r} = \\mu_0(x_0) \\prod_{i=1}^N \\prod_{j=1}^N p_{ij}^{n_ij} $$\n",
    "\n",
    "There is only one observation on $\\mu_0$ so we can't estimate that.  Rather, we focus on estimating $P$.  The MLE is obtained by maximizing the likelihood subject to the constraint that the elements are non-negative and the rows sum to 1.  The solution is\n",
    "\n",
    "$$ \\hat{p}_{ij} = \\frac{n_{ij}}{n_i} $$\n",
    "\n",
    "where $n_i = \\sum_{j=1}^N n_{ij}$.  Here we are assuming that $n_i > 0$.  If not, we set $\\hat{p}_{ij} = 0$ by convention."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 24.30 (Consistency and Asymptotic Normality of the MLE)**.  Assume that the chain is ergodic.  Let $\\hat{p}_{ij}(n)$ denote the MLE after $n$ observations.  Then $\\hat{p}_{ij}(n) \\xrightarrow{\\text{P}} p_{ij}$.  Also,\n",
    "\n",
    "$$ \\left[ \\sqrt{N_i(n)} (\\hat{p}_{ij} - p_{ij})  \\right] \\leadsto N(0, \\Sigma) $$\n",
    "\n",
    "where the left hand side is a matrix, $ N_i(n) = \\sum_{r=1}^n I(X_r = i)$ is the count of observations at state $i$ up to time $n$, and the covariance matrix $\\Sigma$ is a $t \\times t$ matrix, where $t$ is the number of transitions from state $i$ to $j$, with elements\n",
    "\n",
    "$$ \n",
    "\\Sigma_{ij, k\\ell} = \\begin{cases}\n",
    "p_{ij}(1 - p_{ij}) &\\text{if } (i, j) = (k, \\ell) \\\\\n",
    "-p_{ij} p_{i\\ell} &\\text{if } i = k, j \\neq \\ell \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 24.3 Poisson Process"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As the name suggests, the Poisson process is intimately related to the Poisson distribution.  Let's first review the Poisson distribution.\n",
    "\n",
    "Recall that $X$ has a Poisson distribution with parameter $\\lambda$, written $X \\sim \\text{Poisson}(\\lambda)$, if\n",
    "\n",
    "$$ \\mathbb{P}(X = x) \\equiv p(x; \\lambda) = \\frac{e^{-\\lambda} \\lambda^x}{x!}, \\quad = 0, 1, 2, \\dots $$\n",
    "\n",
    "Also recall that:\n",
    "- $X \\sim \\text{Poisson}(\\lambda)$ distribution has mean $\\mathbb{E}(X) = \\lambda$ and variance $\\mathbb{V}(X) = \\lambda$.\n",
    "- If $X \\sim \\text{Poisson}(\\lambda)$, $Y \\sim \\text{Poisson}(\\nu)$ and $X$ and $Y$ are independent, then $X + Y \\sim \\text{Poisson}(\\lambda + \\nu)$.\n",
    "- If $N \\sim \\text{Poisson}(\\lambda)$ and $Y | N = n \\sim \\text{Binomial}(n, p)$ then the marginal distribution of $Y$ is $Y \\sim \\text{Poisson}(\\lambda p)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we describe the Poisson process.  Imagine that whenever an event occurs you record its timestamp.  Let $X_t$ be the number of events that occured up until time $t$.  Then, $\\{ X_t : t \\in [0, \\infty) \\}$ is a stochastic process with state space $\\mathcal{X} = \\{ 0, 1, 2, \\dots \\}$.  A process of this form is called a **counting process**.\n",
    "\n",
    "In what follows, we will sometimes write $X(t)$ instead of $X_t$.  Also, we will need little-o notation:  write $f(h) = o(h)$ if $f(h) / h \\rightarrow 0$ as $h \\rightarrow 0$.  This means that $f(h)$ is smaller than $h$ when $h$ is close to 0.  For example, $h^2 = o(h)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A **Poisson process** is a stochastic process $\\{ X_t : t \\in [0, \\infty) \\}$ with state space $\\mathcal{X} = \\{ 0, 1, 2, \\dots \\}$  such that\n",
    "\n",
    "1.  $X(0) = 0$\n",
    "2.  For any increasing times $0 = t_0 < t_1 < t_2 < \\dots < t_n$, the count increments\n",
    "\n",
    "$$ X(t_1) - X(t_0), X(t_2) - X(t_1), \\dots, X(t_n) - X(t_{n - 1})$$\n",
    "\n",
    "are independent.\n",
    "\n",
    "3.  There is a function $\\lambda(t)$ such that\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbb{P}\\left(X(t + h) - X(t) = 1\\right) &= \\lambda(t)h + o(h) \\\\\n",
    "\\mathbb{P}\\left(X(t + h) - X(t) \\geq 2\\right) &= o(h)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "We call $\\lambda(t)$ the **intensity function**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 24.32**.  If $X_t$ is a Poisson process with intensity function $\\lambda(t)$, then\n",
    "\n",
    "$$ X(s + t) - X(s) \\sim \\text{Poisson}\\left( m(s + t) - m(s) \\right) $$\n",
    "\n",
    "where\n",
    "\n",
    "$$ m(t) = \\int_0^t \\lambda(s) ds $$\n",
    "\n",
    "In particular, $X(t) \\sim \\text{Poisson}(m(t))$.  Hence, $\\mathbb{E}(X(t)) = m(t)$ and $\\mathbb{V}(X(t)) = m(t)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A Poisson process with constant intensity function $\\lambda(t) \\equiv \\lambda$ for some $\\lambda > 0$ is called a **homogeneous Poisson process** with rate $\\lambda$.  In this case,\n",
    "\n",
    "$$ X(t) \\sim \\text{Poisson}(\\lambda t) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $X(t)$be a homogeneous Poisson process with rate $\\lambda$.  Let $W_n$ be the time at which the $n$-th event occurs and set $W_0 = 0$.  The random variables $W_0, W_1, \\dots$ are called **waiting times**.  Let $S_n = W_{n + 1} - W_n$.  Then $S_0, S_1, \\dots$ are called **sojourn times** or **interarrival times**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 24.34**.  The sojourn times $S_0, S_1, \\dots$ are IID random variables.  Their distribution is exponential with mean $1 / \\lambda$, that is, they have density\n",
    "\n",
    "$$ f(s) = \\lambda e^{-\\lambda s}, \\quad s \\geq 0 $$\n",
    "\n",
    "The waiting time has distribution $W_n \\sim \\text{Gamma}(n, 1 / \\lambda)$, that is, it has density\n",
    "\n",
    "$$ f(w) = \\frac{1}{\\Gamma(n)} \\lambda^n w^{n - 1}e^{-\\lambda t} $$\n",
    "\n",
    "Hence, $\\mathbb{E}(W_n) = n / \\lambda$ and $\\mathbb{V}(W_n) = n / \\lambda^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proof**.  First, we have\n",
    "\n",
    "$$ \\mathbb{P}(S_1 > t) = \\mathbb{P}(X(t) = 0) = e^{-\\lambda t}$$\n",
    "\n",
    "which shows that the CDF for $S_1$ is $1 - e^{-\\lambda t}$.  This shows the result for $S_1$. Now,\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbb{P}(S_2 | t | S_1 = s) &= \\mathbb{P}\\left(\\text{no events in } (s, s+t] | S_1 = s\\right) \\\\\n",
    "&= \\mathbb{P}\\left(\\text{no events in } (s, s+t]\\right) \\quad \\text{(increments are independent)} \\\\\n",
    "&= e^{-\\lambda t}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Hence, $S_2$ has an exponential distribution and is independent of $S_1$.  The result follows by repeating the argument.  The result for $W_n$ follows since a sum of exponentials has a Gamma distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 24.6 Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.1** Let $X_0, X_1, \\dots$ be a Markov chain with states $\\{ 0, 1, 2 \\}$ and transition matrix\n",
    "\n",
    "$$ P = \\begin{bmatrix}\n",
    "0.1 & 0.2 & 0.7 \\\\\n",
    "0.9 & 0.1 & 0.0 \\\\\n",
    "0.1 & 0.8 & 0.1\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "Assume that $\\mu_0 = (0.3, 0.4, 0.3)$.  Find $\\mathbb{P}(X_0 = 0, X_1 = 1, X_2 = 2)$ and $\\mathbb{P}(X_0 = 0, X_1 = 1, X_2 = 1)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "We have:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbb{P}(X_0 = 0, X_1 = 1, X_2 = 2) &= \\mathbb{P}(X_0 = 0) \\mathbb{P}(X_1 = 1 | X_0 = 0) \\mathbb{P}(X_2 = 2 | X_1 = 1) \\\\\n",
    "&= \\mu_0(1) P_{12} P_{23} \\\\\n",
    "& = 0.3 \\cdot 0.2 \\cdot 0.0 \\\\\n",
    "& = 0\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "which can also be seen since there is no probability for a transition from state 1 to state 2 (the corresponding value in the $P$ matrix is zero).\n",
    "\n",
    "We also have:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbb{P}(X_0 = 0, X_1 = 1, X_2 = 2) &= \\mathbb{P}(X_0 = 0) \\mathbb{P}(X_1 = 1 | X_0 = 0) \\mathbb{P}(X_2 = 1 | X_1 = 1) \\\\\n",
    "&= \\mu_0(1) P_{12} P_{22} \\\\\n",
    "& = 0.3 \\cdot 0.2 \\cdot 0.1 \\\\\n",
    "& = 0.006\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.2**.  Let $Y_1, Y_2, \\dots$ be a sequence of iid observations such that $\\mathbb{P}(Y = 0) = 0.1$, $\\mathbb{P}(Y = 1) = 0.3$,  $\\mathbb{P}(Y = 2) = 0.2$, $\\mathbb{P}(Y = 3) = 0.4$.  Let $X_0 = 0$ and let\n",
    "\n",
    "$$ X_n = \\max \\{ Y_1, \\dots, Y_n \\} $$\n",
    "\n",
    "Show that $X_0, X_1, \\dots$ is a Markov chain and find the transition matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  By definition,\n",
    "\n",
    "$$ X_{n + 1} = \\max \\{Y_1, \\dots, Y_{n+1} \\} = \\max \\{ X_n, Y_{n + 1} \\} $$\n",
    "\n",
    "so $X_{n + 1}$ is defined based only on its predecessor and on a IID variable $Y_{n+1} \\sim Y$.  Thus:\n",
    "\n",
    "$$ \\mathbb{P}(X_{n + 1} = j | X_n = i) = \\begin{cases}\n",
    "\\frac{\\mathbb{P}(Y = j)}{\\sum_{k \\geq i} \\mathbb{P}(Y = k)} & \\text{if } j \\geq i \\\\\n",
    "0 &\\text{if } j < i\n",
    "\\end{cases} $$\n",
    "\n",
    "This, paired with a state space $\\mathcal{X} = \\{ 0, 1, 2, 3 \\} $ and initial probabilities $\\mu_0 = (1, 0, 0, 0)$, defines a Markov chain for the $X_i$'s.  The explicit transition matrix is:\n",
    "\n",
    "$$ P = \\begin{bmatrix}\n",
    "1/10 & 3/10 & 1/5 & 2/5 \\\\\n",
    "0   & 1/3 & 2/9 & 4/9 \\\\\n",
    "0   & 0   & 1/3 & 2/3 \\\\\n",
    "0   & 0   & 0   & 1\n",
    "\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.3**.  Consider a two state Markov chain with states $\\mathcal{X} = \\{ 1, 2 \\}$ and transition matrix\n",
    "\n",
    "$$ P = \\begin{bmatrix}\n",
    "1 - a & a \\\\\n",
    "b & 1 - b\n",
    "\\end{bmatrix} $$\n",
    "\n",
    "where $0 < a < 1$ and $0 < b < 1$.  Prove that\n",
    "\n",
    "$$ \\lim_{n \\rightarrow \\infty} P^n = \\begin{bmatrix}\n",
    "\\frac{b}{a + b} & \\frac{a}{a + b} \\\\\n",
    "\\frac{b}{a + b} & \\frac{a}{a + b}\n",
    "\\end{bmatrix} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  Note that the Markov chain is irreducible and ergodic, given the bounds on $a$ and $b$.  Note also that it has a stationary distribution\n",
    "\n",
    "$$ \\pi = \\left( \\frac{b}{a + b}, \\frac{a}{a + b} \\right) $$\n",
    "\n",
    "since $\\pi = \\pi P$:\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "\\pi P &= \n",
    "\\begin{bmatrix} \n",
    "\\frac{b}{a + b} & \\frac{a}{a + b}\n",
    "\\end{bmatrix} \\begin{bmatrix}\n",
    "1 - a & a \\\\\n",
    "b & 1 - b\n",
    "\\end{bmatrix} \\\\\n",
    "&= \\frac{1}{a + b} \\begin{bmatrix}\n",
    " b (1 - a) + ab &\n",
    " ab + (1 - b) a \n",
    "\\end{bmatrix} \\\\\n",
    "&= \\begin{bmatrix} \\frac{b}{a + b} & \\frac{a}{a + b} \\end{bmatrix}\\\\\n",
    "&= \\pi\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "By theorem 24.24, the limit of $P^n$ is as given,\n",
    "\n",
    "$$ \\lim_{n \\rightarrow \\infty} P^n = \n",
    "\\begin{bmatrix}\n",
    "\\pi \\\\ \\pi\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "\\frac{b}{a + b} & \\frac{a}{a + b} \\\\\n",
    "\\frac{b}{a + b} & \\frac{a}{a + b}\n",
    "\\end{bmatrix} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.4**.  Consider the chain from question 3 and set $a = .1$ and $b = .3$.  Simulate the chain.  Let\n",
    "\n",
    "$$\n",
    "\\hat{p}_n(1) = \\frac{1}{n} \\sum_{i=1}^n I(X_i = 1)\n",
    "\\quad \\text{and} \\quad\n",
    "\\hat{p}_n(2) = \\frac{1}{n} \\sum_{i=1}^n I(X_i = 2)\n",
    "$$\n",
    "\n",
    "be the proportion of times the chain is in state 1 and state 2.  Plot $\\hat{p}_n(1)$ and $\\hat{p}_n(2)$ versus $n$ and verify that they converge to the values predicted from the answer in the previous question."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "a, b = 0.1, 0.3\n",
    "P = np.array([[1 - a, a], [b, 1 - b]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Do a *single* simulation starting from, say, state 1\n",
    "\n",
    "def generate_series(n, seed=None, initial_state=1):\n",
    "    if seed is not None:\n",
    "        np.random.seed(0)\n",
    "\n",
    "    random_values = np.random.uniform(low=0, high=1, size=n)\n",
    "\n",
    "    X = np.empty(n, dtype=int)\n",
    "    X[0] = initial_state\n",
    "    for i in range(1, n):\n",
    "        X[i] = 1 if random_values[i] < P[X[i - 1] - 1, 0] else 2\n",
    "        \n",
    "    return X\n",
    "\n",
    "n = 10000\n",
    "X = generate_series(n, seed=0)\n",
    "p1 = np.cumsum(X == 1) / np.arange(1, n + 1)\n",
    "p2 = np.cumsum(X == 2) / np.arange(1, n + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "plt.plot(np.arange(1, n+1), p1, label=r'$\\hat{p}_n(1)$')\n",
    "plt.plot(np.arange(1, n+1), p2, label=r'$\\hat{p}_n(2)$')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the values do converge to $\\pi = (0.75, 0.25)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.5**.  An important Markov chain is the **branching process** which is used in biology, genetics, nuclear physics and many other fields.  Suppose that an animal has $Y$ children.  Let $p_k = \\mathbb{P}(Y = k)$.  Hence $p_k \\geq 0$ for all $k$ and $\\sum_{k = 0}^\\infty p_k = 1$.  Assume each animal has the same lifespan and that they produce offspring according to the distribution $p_k$.  Let $X_n$ be the number of animals in the $n$-th generation.  Let $Y_1^{(n)}, \\dots, Y_{X_n}^{(n)}$ be the offspring produced in the $n$-th generation.  Note that\n",
    "\n",
    "$$ X_{n+1} = Y_1^{(n)} + \\dots + Y_{X_n}^{(n)}$$\n",
    "\n",
    "Let $\\mu = \\mathbb{E}(Y)$ and $\\sigma^2 = \\mathbb{V}(Y)$.  Assume throughout this question that $X_0 = 1$.  Let $M(n) = \\mathbb{E}(X_n)$ and $V(n) = \\mathbb{V}(X_n)$.\n",
    "\n",
    "**(a)**  Show that $M(n + 1) = \\mu M(n)$ and that $V(n + 1) = \\sigma^2 M(n) + \\mu^2 V(n) $.\n",
    "\n",
    "**(b)**  Show that $M(n) = \\mu^n$ and that $V(n) = \\sigma^2 \\mu^{n-1} (1 + \\mu + \\dots + \\mu^{n - 1})$.\n",
    "\n",
    "**(c)**  What happens to the variance if $\\mu > 1$?  What happens to the variance if $\\mu = 1$?  What happens to the variance if $\\mu < 1$?\n",
    "\n",
    "**(d)**  The population goes extinct if $X_n = 0$ for some $n$.  Let us thus define the extinction time $N$ by\n",
    "\n",
    "$$ N = \\min \\{ n : X_n = 0 \\} $$\n",
    "\n",
    "Let $F(n) = \\mathbb{P}(N \\leq n)$ be the CDF of the random variable $N$.  Show that\n",
    "\n",
    "$$ F(n) = \\sum_{k=0}^\\infty p_k \\left( F(n - 1) \\right)^k, \\quad n = 1, 2, \\dots $$\n",
    "\n",
    "Hint: note that the event $\\{ N \\leq n \\}$ is the same event as $\\{ X_n = 0 \\}$.  Thus $\\mathbb{P}(N \\leq n) = \\mathbb{P}(X_n = 0)$.  Let $k$ be the number of offspring of the original event.  The population becomes extinct at time $n$ if and only if each of the $k$ sub-populations generated from the $k$ offspring goes extinct in $n - 1$ generations.\n",
    "\n",
    "**(e)**  Suppose that $p_0 = 1/4$, $p_1 = 1/2$, $p_2 = 1/4$.  Use the formula from (d) to compute the CDF $F(n)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(a)**  We have:\n",
    "\n",
    "$$ M(n + 1) = \\mathbb{E}[X_{n + 1}] = \\mathbb{E}\\left[ \\sum_{i = 1}^{X_n} Y_i^{(n)} \\right] = \\mathbb{E}\\left[ \\sum_{i = 1}^{X_n} \\mathbb{E}[Y_i^{(n)}] \\right] = \\mathbb{E}\\left[ \\sum_{i = 1}^{X_n} \\mu \\right]  = \\mu \\mathbb{E}[X_n] = \\mu M(n)$$\n",
    "\n",
    "We also have:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "V(n + 1) &= \\mathbb{V}[X_{n + 1}] = \\mathbb{E}[X_{n + 1}^2] - \\mathbb{E}[X_{n + 1}]^2 \\\\\n",
    "&= \\mathbb{E}\\left[\\left( \\sum_{i = 1}^{X_n} Y_i^{(n)} \\right)^2\\right] - \\mu^2 M(n)^2 \\\\\n",
    "&= \\mathbb{E}\\left[ \\sum_{i = 1}^{X_n} \\left( Y_i^{(n)} \\right)^2 + \\sum_{i = 1}^{X_n} \\sum_{j = 1, j \\neq i}^{X_n} Y_i^{(n)} Y_j^{(n)} \\right] - \\mu^2 M(n)^2 \\\\\n",
    "&= \\mathbb{E}\\left[ \\sum_{i = 1}^{X_n} \\mathbb{E}\\left[\\left( Y_i^{(n)} \\right)^2\\right] + \\sum_{i = 1}^{X_n} \\sum_{j = 1, j \\neq i}^{X_n} \\mathbb{E}\\left[ Y_i^{(n)} Y_j^{(n)} \\right] \\right] - \\mu^2 M(n)^2 \\\\\n",
    "&= \\mathbb{E}\\left[ \\sum_{i = 1}^{X_n} \\left( \\mathbb{V}[Y_i^{(n)}] +  \\mathbb{E}\\left[ Y_i^{(n)} \\right]^2 \\right) + \\sum_{i = 1}^{X_n} \\sum_{j = 1, j \\neq i}^{X_n} \\mathbb{E}\\left[ Y_i^{(n)} \\right] \\mathbb{E} \\left[ Y_j^{(n)} \\right] \\right] - \\mu^2 M(n)^2 \\\\\n",
    "&= \\mathbb{E}\\left[ \\sum_{i = 1}^{X_n} \\left( \\sigma^2 + \\mu^2 \\right) + \\sum_{i = 1}^{X_n} \\sum_{j = 1, j \\neq i}^{X_n} \\mu^2 \\right] - \\mu^2 M(n)^2 \\\\\n",
    "&= \\mathbb{E} \\left[ X_n (\\sigma^2 + \\mu^2) + X_n (X_n - 1) \\mu^2 \\right] - \\mu^2 M(n)^2 \\\\\n",
    "&= \\mathbb{E} \\left[ X_n \\sigma^2 + X_n^2 \\mu^2 \\right] - \\mu^2 M(n)^2 \\\\\n",
    "&= \\sigma^2 \\mathbb{E} [ X_n ] + \\mu^2 \\mathbb{E} [ X_n^2 ] - \\mu^2 M(n)^2 \\\\\n",
    "&= \\sigma^2 M(n) + \\mu^2 (V(n) + M(n)^2) - \\mu^2 M(n)^2 \\\\\n",
    "&= \\sigma^2 M(n) + \\mu^2 V(n)\n",
    "\\end{align}\n",
    "$$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(b)**\n",
    "\n",
    "Since the initial population is 1, $M(0) = 1$; since $M(n + 1) = \\mu M(n)$, it follows by induction that $M(n) = \\mu^n$.\n",
    "\n",
    "Now, we also have that the initial population is known, so $V(0) = 0$. By induction,\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "V(n + 1) &= \\sigma^2 M(n) + \\mu^2 V(n) \\\\\n",
    "&= \\sigma^2 \\mu^n + \\mu^2 \\left( \\sigma^2 \\mu^{n - 1} \\left( 1 + \\mu + \\dots + \\mu^{n - 1}\\right) \\right) \\\\\n",
    "&= \\sigma^2 \\mu^n \\left(1 + \\mu \\left( 1 + \\mu + \\dots + \\mu^{n - 1} \\right) \\right) \\\\\n",
    "&= \\sigma^2 \\mu^n \\left(1 + \\mu + \\dots + \\mu^n \\right)\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(c)**  If $\\mu \\neq 1$, we can add up the geometric sum and write\n",
    "\n",
    "$$ V(n) = \\sigma^2 \\mu^n \\frac{1 - \\mu^{n}}{1 - \\mu}$$\n",
    "\n",
    "- For $\\mu > 1$, $V(n)$ grows exponentially.  \n",
    "- For $\\mu < 1$, $V(n)$ converges to 0.  \n",
    "- For $\\mu = 1$, we have $V(n) = n \\sigma^2$, which grows linearly."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(d)**  Following the reasoning in the hint,\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "F(n) &= \\mathbb{P}(N \\leq n) = \\mathbb{P}(X_n = 0) \\\\\n",
    "&= \\sum_{k=0}^\\infty \\mathbb{P}(X_n = 0 | X_1 = k) \\mathbb{P}(X_1 = k) \\\\\n",
    "&= \\sum_{k=0}^\\infty \\mathbb{P}(Y_i^{(n - 1)} = 0 \\text{ for all } i | X_1 = k) \\mathbb{P}(X_1 = k) \\\\\n",
    "&= \\sum_{k=0}^\\infty \\left( \\prod_{i=1}^k \\mathbb{P}(Y_i^{(n - 1)} = 0 | X_1 = k) \\right) \\mathbb{P}(X_1 = k) \\\\\n",
    "&= \\sum_{k=0}^\\infty \\left( \\prod_{i=1}^k F(n-1) \\right) p_k \\\\\n",
    "&= \\sum_{k=0}^\\infty p_k \\left( F(n - 1)\\right)^k\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(e)**  The recurrence formula is:\n",
    "\n",
    "$$ F(n + 1) = p_0 + p_1 F(n) + p_2 F(n)^2 = \\frac{1}{4} + \\frac{1}{2} F(n) + \\frac{1}{4} F(n)^2 = \\frac{1}{4} (F(n) + 1)^2$$\n",
    "\n",
    "with initial value $F(0) = 0$ (since we start with a non-extinct population)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "N = 1000\n",
    "F = np.empty(N)\n",
    "\n",
    "F[0] = 0\n",
    "for n in range(1, N):\n",
    "    F[n] = (F[n - 1] + 1)**2 / 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "plt.plot(np.arange(1, N + 1), F)\n",
    "plt.xlabel('N')\n",
    "plt.ylabel('F(N)')\n",
    "plt.xscale('log')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.6**. Let\n",
    "\n",
    "$$ P = \\begin{bmatrix}\n",
    "0.40 & 0.50 & 0.10 \\\\\n",
    "0.05 & 0.70 & 0.25 \\\\\n",
    "0.05 & 0.50 & 0.45\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "Find the stationary distribution $\\pi$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  The stationary distribution $\\pi$  satisfies $\\pi = \\pi P$, so it is a (normalized) left eigenvector of $P$, or a right eigenvector of $P^T$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy.linalg import eig\n",
    "\n",
    "P = np.array([[0.4, 0.5, 0.1], [0.05, 0.7, 0.25], [0.05, 0.5, 0.45]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.11041049 0.89708523 0.42784065]\n"
     ]
    }
   ],
   "source": [
    "w, v = eig(P.T)\n",
    "pi = v[:, 0]\n",
    "print(pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.11041049 0.89708523 0.42784065]\n"
     ]
    }
   ],
   "source": [
    "print(pi @ P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.7**.  Show that if $i$ is a recurrent state and $i \\leftrightarrow j$, then $j$ is a recurrent state."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "Since $i$ is recurrent, $\\sum_n p_{ii}(n) = \\infty$.  But \n",
    "\n",
    "$$\\sum_n p_{jj}(n) \\geq \\sum_{a, b, c} p_{ji}(a) p_{ii}(b) p_{ij}(c) \\geq \\sum_b p_{ji}(a') p_{ii}(b) p_{ij}(c')$$\n",
    "\n",
    "for some $a'$ such that $p_{ji}(a') > 0$ (which exists because $j \\rightarrow i$) and some $c'$ such that $p_{ij}(c') > 0$ (which exists because $i \\rightarrow j$).  Then this sum is lower bounded by \n",
    "\n",
    "$$\\frac{1}{p_{ji}(a') p_{ij}(c')} \\sum_b p_{ii}(b)$$\n",
    "\n",
    "which diverges because $\\sum_n p_{ii}(n) = \\infty$, so $j$ must be a recurrent state."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.8**.  Let\n",
    "\n",
    "$$ P = \\begin{bmatrix}\n",
    "1/3 & 0   & 1/3 & 0 & 0 & 1/3 \\\\\n",
    "1/2 & 1/4 & 1/4 & 0 & 0 & 0   \\\\\n",
    "0   & 0   & 0   & 0 & 1 & 0   \\\\\n",
    "1/4 & 1/4 & 1/4 & 0 & 0 & 1/4 \\\\\n",
    "0   & 0  & 1    & 0 & 0 & 0   \\\\\n",
    "0   & 0  & 0    & 0 & 0 & 1\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "Which states are transient?  Which states are recurrent?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "from graphviz import Digraph\n",
    "\n",
    "d = Digraph()\n",
    "\n",
    "d.edge('1', '1', label='1/3')\n",
    "d.edge('1', '3', label='1/3')\n",
    "d.edge('1', '6', label='1/3')\n",
    "\n",
    "d.edge('2', '1', label='1/2')\n",
    "d.edge('2', '2', label='1/4')\n",
    "d.edge('2', '3', label='1/4')\n",
    "\n",
    "d.edge('3', '5', label='1')\n",
    "\n",
    "d.edge('4', '1', label='1/4')\n",
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    "d.edge('4', '6', label='1/4')\n",
    "\n",
    "d.edge('5', '3', label='1')\n",
    "\n",
    "d.edge('6', '6', label='1')\n",
    "\n",
    "d"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "States 3, 5, 6 are recurrent:\n",
    "\n",
    "- A chain starting on state 6 will forever stay in state 6.\n",
    "- A chain starting on states 3 or 5 will bounce between states 3 and 5 forever.\n",
    "\n",
    "Other states are not recurrent, as they may eventually fall into state 6 and never reach other state after."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.9**.  Let\n",
    "\n",
    "$$ P = \\begin{bmatrix}\n",
    "0 & 1 \\\\\n",
    "1 & 0\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "Show that $\\pi = (1/2, 1/2)$ is a stationary distribution.  Does this chain converge?  Why / why not?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  The distribution is stationary if and only if $\\pi = \\pi P$, and \n",
    "\n",
    "$$ \\pi P = \\begin{bmatrix} 1/2 & 1/2 \\end{bmatrix}\n",
    "\\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}\n",
    "= \\begin{bmatrix} 1/2 & 1/2 \\end{bmatrix} = \\pi\n",
    "$$\n",
    "\n",
    "This chain does not converge: it will always oscillate between states 1 and 2 with probability 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.10**.  Let $0 < p < 1$ and $q = 1 - p$.  Let\n",
    "\n",
    "$$ P = \n",
    "\\begin{bmatrix}\n",
    "q & p & 0 & 0 & 0 \\\\\n",
    "q & 0 & p & 0 & 0 \\\\\n",
    "q & 0 & 0 & p & 0 \\\\\n",
    "q & 0 & 0 & 0 & p \\\\\n",
    "1 & 0 & 0 & 0 & 0\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Find the limiting distribution of the chain."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solving $\\pi P = \\pi$, we get:\n",
    "\n",
    "$$ \\pi = \\begin{bmatrix} \n",
    "\\frac{1 - p}{p^4(1 - p^5)} &\n",
    "\\frac{1 - p}{p^3(1 - p^5)} &\n",
    "\\frac{1 - p}{p^2(1 - p^5)} &\n",
    "\\frac{1 - p}{p(1 - p^5)} &\n",
    "\\frac{1 - p}{1 - p^5}\n",
    "\\end{bmatrix} $$\n",
    "\n",
    "We can verify this is the limiting distribution:\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "\\pi P &= \\begin{bmatrix} \\frac{1 - p}{p^4(1 - p^5)} &\n",
    "\\frac{1 - p}{p^3(1 - p^5)} &\n",
    "\\frac{1 - p}{p^2(1 - p^5)} &\n",
    "\\frac{1 - p}{p(1 - p^5)} &\n",
    "\\frac{1 - p}{1 - p^5}\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "1 - p & p & 0 & 0 & 0 \\\\\n",
    "1 - p & 0 & p & 0 & 0 \\\\\n",
    "1 - p & 0 & 0 & p & 0 \\\\\n",
    "1 - p & 0 & 0 & 0 & p \\\\\n",
    "1 & 0 & 0 & 0 & 0\n",
    "\\end{bmatrix} \\\\\n",
    "&= \\begin{bmatrix} \n",
    "\\frac{(1 - p)^2}{p^4(1 - p^5)}\n",
    "+ \\frac{(1 - p)^2}{p^3(1 - p^5)}\n",
    "+ \\frac{(1 - p)^2}{p^2(1 - p^5)}\n",
    "+ \\frac{(1 - p)^2}{p(1 - p^5)}\n",
    "+ \\frac{1 - p}{1 - p^5} &\n",
    "\\frac{1 - p}{p^3(1 - p^5)} &\n",
    "\\frac{1 - p}{p^2(1 - p^5)} &\n",
    "\\frac{1 - p}{p(1 - p^5)} &\n",
    "\\frac{1 - p}{1 - p^5}\n",
    "\\end{bmatrix} \\\\\n",
    "&= \\begin{bmatrix} \n",
    "\\frac{(1 - p)^2(1 + p + p^2 + p^3) + p^4(1 - p)}{p^4(1 - p^5)} &\n",
    "\\frac{1 - p}{p^3(1 - p^5)} &\n",
    "\\frac{1 - p}{p^2(1 - p^5)} &\n",
    "\\frac{1 - p}{p(1 - p^5)} &\n",
    "\\frac{1 - p}{1 - p^5}\n",
    "\\end{bmatrix} \\\\\n",
    "&= \\begin{bmatrix} \n",
    "\\frac{1 - p}{p^4(1 - p^5)} &\n",
    "\\frac{1 - p}{p^3(1 - p^5)} &\n",
    "\\frac{1 - p}{p^2(1 - p^5)} &\n",
    "\\frac{1 - p}{p(1 - p^5)} &\n",
    "\\frac{1 - p}{1 - p^5}\n",
    "\\end{bmatrix} \\\\\n",
    "&= \\pi\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.11**.  Let $X(t)$ be an inhomogeneous Poisson process with intensity function $\\lambda(t) > 0$.  Let $\\Lambda(t) = \\int_0^t \\lambda(u) du$.  Define $Y(s) = X(t)$ where $s = \\Lambda(t)$.  Show that $Y(s)$ is a homogeneous Poisson process with intensity $\\lambda = 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "We have:  $Y(s) = X(\\Lambda^{-1}(s))$ and $X(t) \\sim \\text{Poisson}(\\Lambda(t))$, so \n",
    "\n",
    "$$Y(s) \\sim \\text{Poisson}(\\Lambda(\\Lambda^{-1}(s)) = \\text{Poisson}(s) = \\text{Poisson}(\\lambda_Y \\cdot s)$$\n",
    "\n",
    "which is the desired result with $\\lambda_Y = 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.12**.  Let $X(t)$ be a Poisson process with intensity $\\lambda$.  Find the conditional distribution of $X(t)$ given that $X(t + s) = n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  The random variables $X(t) - X(0) = X(t)$ and $Y(s) = X(t + s) - X(t)$ are independent, as they are both count increments, with $X(t) \\sim \\text{Poisson}(\\lambda t)$ and $Y(s) \\sim \\text{Poisson}(\\lambda s)$. Then:\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "\\mathbb{P}\\left(X(t) = x | X(t + s) = n\\right) &= \\mathbb{P}\\left(X(t) = x | X(t) + Y(s) = n\\right) \\\\\n",
    "&= \\frac{\\mathbb{P}\\left(X(t) = x, X(t) + Y(s) = n\\right)}{\\mathbb{P}\\left(X(t) + Y(s) = n\\right)} \\\\\n",
    "&= \\frac{\\mathbb{P}\\left(X(t) = x, Y(s) = n - x\\right)}{\\sum_{0 \\leq j \\leq n} \\mathbb{P}\\left(X(t) = j, Y(s) = n - j\\right)} \\\\\n",
    "&= \\frac{\\mathbb{P}\\left(X(t) = x\\right) \\mathbb{P}\\left(Y(s) = n - x\\right)}{\\sum_{j=0}^n \\mathbb{P}\\left(X(t) = j\\right) \\mathbb{P}\\left( Y(s) = n - j\\right)} \\\\\n",
    "&= \\frac{f(x)}{\\sum_{j=0}^n f(j) }\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "where $f(x) = \\mathbb{P}\\left(X(t) = x\\right) \\mathbb{P}\\left( Y(s) = n - x\\right)$.  But we have:\n",
    "\n",
    "$$ f(x) = \\frac{(\\lambda t)^x e^{-\\lambda t}}{x!}\\frac{(\\lambda s)^{n - x} e^{-\\lambda s}}{(n - x)!} = \\frac{\\lambda^n e^{-\\lambda (t + s)}}{n!} \\binom{n}{x} t^x s^{n - x} $$\n",
    "\n",
    "Replacing on the expression above and cancelling the factors that do not depend on $x$, we get\n",
    "\n",
    "$$ \\mathbb{P}\\left(X(t) = x | X(t + s) = n\\right) = \\frac{f(x)}{\\sum_{j=0}^n f(j) } = \\frac{\\binom{n}{x} t^x s^{n - x}}{\\sum_{j=0}^n \\binom{n}{j} t^j s^{n - j}} \n",
    "= \\frac{\\binom{n}{x} t^x s^{n - x}}{(t + s)^n} = \\binom{n}{x} \\left( \\frac{t}{t + s} \\right)^x \\left( \\frac{s}{t + s}\\right)^{n - x}$$\n",
    "\n",
    "using the binomial expansion $(t + s)^n = \\sum_{j=0}^n \\binom{n}{j} t^j s^{n - j}$.  Therefore, the conditional distribution follows a Binomial distribution,\n",
    "\n",
    "$$ X(t) | X(t + s) = n \\sim \\text{Binomial}\\left(n, \\frac{t}{t + s} \\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.13**.  Let $X$ be a Poisson process with intensity $\\lambda$.  Find the probability that $X(t)$ is odd, i.e. $\\mathbb{P}(X(t) = 1, 3, 5, \\dots)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  Expanding using the probability mass function,\n",
    "\n",
    "$$\n",
    "\\mathbb{P}(X(t) \\text{ is odd}) = \\sum_{k=0}^\\infty \\frac{(\\lambda t)^{2k + 1} e^{-\\lambda t}}{(2k + 1)!} = e^{-\\lambda t} \\text{sinh} (\\lambda t) = \\frac{1}{2}\\left( 1 - e^{-2 \\lambda t}\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.14**.  Suppose that people logging in to the University computer system is described by a Poisson process $X(t)$ with intensity $\\lambda$.  Assume that a person stays logged in for some random time with CDF $G$. Assume these times are all independent.  Let $Y(t)$ be the number of people on the system at time $t$.  Find the distribution of $Y(t)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "The number of people arriving at time period $j$ is $A(j) \\equiv X(j) - X(j-1)$, where $X(0) = 0$.  Let $W_i^{(j)}$ be the amount of time spent by the $i$-th person that arrived at time $j$.  Then, the total number of people at time $t$ is:\n",
    "\n",
    "$$ Y(t) = \\sum_{j=1}^t \\sum_{i=1}^{A(j)} I\\left(W_i^{(j)} \\geq t - j\\right) $$\n",
    "\n",
    "where $I(x) = 1$ if $x$ is true and $0$ otherwise, and the terms count only the people who are still logged in after an extra time $t - j$ elapsed.\n",
    "\n",
    "Each value $I\\left(W_i^{(j)} \\geq x\\right)$ follows a distribution given by $1 - G(x)$, measuring the probability of a person staying longer than time $x$, and they are all independent.\n",
    "\n",
    "We can write the sum above as:\n",
    "\n",
    "$$ Y(t) = \\sum_{j=1}^t B(j), \\quad B(j) =\\sum_{i = 1}^{A(j)} I\\left(W_i^{(j)} \\geq t - j\\right) $$\n",
    "\n",
    "Note that this implies that $B(j) | A(j) = n$ follows a binomial distribution with parameters $n$ and $\\mathbb{P}\\left(W_i^{(j)} \\geq t - j \\right) = 1 - G(t - j)$.\n",
    "\n",
    "Since $A(j) \\sim \\text{Poisson}(\\lambda)$ and  $B(j) | A(j) = n \\sim \\text{Binomial}(n, 1 - G(t - j))$, the marginal distribution of $B(j)$ is $B(j) \\sim \\text{Poisson}\\left(\\lambda (1 - G(t - j))\\right)$. Therefore, $Y$ being the sum of independent Poisson variables $B(j)$, we get that $Y$ is a Poisson process with inhomogeneous rate $\\lambda_Y(t)$,\n",
    "\n",
    "$$ Y(t) \\sim \\text{Poisson}\\left( \\lambda_Y(t) \\right), \\quad \\lambda_Y(t) = \\lambda \\left[ \\sum_{j=1}^t(1 - G(t - j)) \\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.15**.  Let $X(t)$ be a Poisson process with intensity $\\lambda$.  Let $W_1, W_2, \\dots$ be the waiting times.  Let $f$ be an arbitrary function.  Show that\n",
    "\n",
    "$$ \\mathbb{E} \\left[ \\sum_{i=1}^{X(t)} f(W_i) \\right] = \\lambda \\int_0^t f(w) dw $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "When conditioning on $X(t) = n$, the waiting times are uniformly distributed on the interval $[0, t]$, so $\\sum_{i=1}^n f(W_i)$ has the same distribution as  $\\sum_{i=1}^n f\\left(U_i^{(n)}\\right)$, for $U_i^{(n)} \\sim \\text{Uniform}(0, t)$.  Then the expectation of $f\\left(U_i^{(n)}\\right)$ is \n",
    "\n",
    "$$\\mathbb{E}\\left[ f\\left(U_i^{(n)}\\right) \\right] = \\int_0^t f(w) f_{U_i^{(n)}}(w) dw = \\frac{1}{t} \\int_0^t f(w) dw$$\n",
    "\n",
    "Then, we have:\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "\\mathbb{E}\\left[ \\sum_{i=1}^{X(t)} f(W_i) \\right] \n",
    "&= \\sum_{n = 0}^\\infty \\mathbb{E}\\left[ \\sum_{i=1}^{X(t)} f(W_i) \\; \\Bigg| \\; X(t) = n\\right] \\mathbb{P} \\left[ X(t) = n \\right] \\\\\n",
    "&= \\sum_{n = 0}^\\infty \\mathbb{E}\\left[ \\sum_{i=1}^n f\\left(U_i^{(n)}\\right) \\right] \\mathbb{P} \\left[ X(t) = n \\right] \\\\\n",
    "&= \\sum_{n = 0}^\\infty \\frac{n}{t} \\left( \\int_0^t f(w) dw \\right) \\mathbb{P} \\left[ X(t) = n \\right] \\\\\n",
    "&= \\frac{1}{t} \\left( \\int_0^t f(w) dw \\right) \\sum_{n=0}^\\infty n \\; \\mathbb{P} \\left[ X(t) = n \\right] \\\\\n",
    "&= \\frac{1}{t} \\left( \\int_0^t f(w) dw \\right) \\mathbb{E} \\left[ X(t) \\right] \\\\\n",
    "&= \\frac{1}{t} \\left( \\int_0^t f(w) dw \\right) \\lambda t \\\\\n",
    "&= \\lambda \\int_0^t f(w) dw\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 24.6.16**.  A two dimensional Poisson process is a process of random points in the plane such that (i) for any set $A$, the number of points falling in $A$ is Poisson with mean $\\lambda \\mu(A)$ where $\\mu(A)$ is the area of $A$, (ii) the number of points in nonoverlapping regions is independent.  Consider an arbitrary point $x_0$ in the plane.  Let $X$ denote the distance from $x_0$ to the nearest point.  Show that\n",
    "\n",
    "$$ \\mathbb{P}(X > t) = e^{-\\lambda \\pi t^2} $$\n",
    "\n",
    "and\n",
    "\n",
    "$$ \\mathbb{E}(X) = \\frac{1}{2 \\sqrt{\\lambda}} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  The distance $X$ from $x_0$ to the nearest point is at least $t$ if and only if no points lie within a circle of radius $t$ centered in $x_0$.  This circle $C$ has area $\\mu(C) = \\pi t^2$, and so\n",
    "\n",
    "$$ \\mathbb{P}(X > t) = \\mathbb{P}(\\# \\{\\text{points in C} \\} = 0) = f_C(0) = \\frac{\\lambda_C^0 e^{-\\lambda_C}}{0!} = e^{-\\lambda_C} = e^{-\\lambda \\pi t^2} $$\n",
    "\n",
    "Given that $X$ has a CDF of $F_X(x) = 1 - e^{-\\lambda \\pi x^2}$ for $x \\geq 0$, its PDF is $f_X(x) = 2 \\lambda \\pi x e^{-\\lambda \\pi x^2}$, and its mean is\n",
    "\n",
    "$$ \\mathbb{E}(X) = \\int_0^\\infty x f_X(x) \\; dx  = \\int_0^\\infty 2 \\lambda \\pi x^2 e^{-2 \\lambda \\pi x^2} \\; dx = \\frac{1}{2 \\sqrt{\\lambda}}$$"
   ]
  }
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