Add content for functions/integrals-prob-density-funcs

Add actual content to the skeleton of the
`functions/integrals-prob-density-funcs` section.

Signed-off-by: Eggert Karl Hafsteinsson <[email protected]>
Signed-off-by: Teodor Dutu <[email protected]>
Signed-off-by: Razvan Deaconescu <[email protected]>

chapters/functions/integrals-prob-density-funcs/reading/README.md

@@ -1 +1,245 @@
 # Integrals and Probability Density Functions
+
+## Area Under a Curve
+
+The area under a curve between $x=a$ and $x=b$ (for a positive function) is called the integral of the function.
+
+![Fig. 28](../media/16_1_Area_under_a_curve.png)
+
+Figure: Example 1, 2 and 3
+
+### Details
+
+:::note Definition
+
+The area under a curve between $x=a$ and $x=b$ (for a positive function) is called the **integral of the function** and is denoted: $\int_{a}^{b} f(x)dx$ when it exists.
+
+:::
+
+## The Antiderivative
+
+Given a function $f$, if there is another function $F$ such that $F'=f$, we say that $F$ is the **antiderivative** of $f$.
+For a function $f$ the antiderivative is denoted by $\int f dx$.
+
+Note that if $F$ is one antiderivative of $f$ and $C$ is a constant, then $G=F+C$ is also an antiderivative.
+It is therefore customary to always include the constant, e.g. $\int x dx=\displaystyle\frac{1}{2}x^2+C$.
+
+### Examples
+
+:::info Example
+
+The antiderivative of $x$ to a power raises the power by one and divides it by the new power:
+
+$$\int x^n dx=\displaystyle\frac{1}{n+1}x^{n+1} +C$$
+
+:::
+
+:::info Example
+
+$\int e^x dx=e^x+C$.
+
+:::
+
+:::info Example
+
+$\int \frac{1}{x} dx=\ln(x)+C$.
+
+:::
+
+:::info Example
+
+$\int 2xe^{x^2} dx=e^{x^2}+C$.
+
+:::
+
+## The Fundamental Theorem of Calculus
+
+If $f$ is a continuous function, and $F'(x)=f(x)$ for $x\in[a,b]$, then $\int_a^b f(x)dx=F(b)-F(a)$
+
+### Detail
+
+It is not too hard to see that the area under the graph of a positive function $f$ on the interval $[a,b]$ must be equal to the difference of the values of its antiderivative at $a$ and $b$.
+This also holds for functions which take on negative values and is formally stated below.
+
+:::note Definition
+
+**Fundamental theorem of calculus:** If $F$ is the antiderivative of the continuous function $f$, i.e. $F'=f$ for $x\in[a,b]$, then $\int_a^b f(x)dx=F(b)-F(a)$.
+
+This difference is often written as $\int_a^b f dx$ or $[F(x)]_a ^b$.
+
+:::
+
+### Examples
+
+:::info Example
+
+The area under the graph of $x^n$ between $0$ and $3$ is $\int_0^3 x^n dx = [\displaystyle\frac{1}{n+1}x^{n+1}]_0 ^3=\displaystyle\frac{1}{n+1}3^{n+1}-\displaystyle\frac{1}{n+1}0^{n+1}=\displaystyle\frac{3^{n+1}}{n+1}$
+
+:::
+
+:::info Example
+
+The area under the graph of $e^x$ between $3$ and $4$ is $\int_3^4 e^x dx =[e^x]_3 ^4= e^4-e^3$
+
+:::
+
+:::info Example
+
+The area under the graph of $\displaystyle\frac{1}{x}$ between $1$ and $a$ is $\int_1^a \frac{1}{x} dx =[\ln(x)]_1 ^a= \ln(a)-\ln(1)=\ln(a)$
+
+:::
+
+## Density Functions
+
+The probability density function (`p.d.f.`) and the cumulative distribution function (`c.d.f.`).
+
+![Fig. 29](../media/16_4_Density_functions.png)
+
+### Details
+
+:::note Definition
+
+If $X$ is a random variable such that
+
+$$P(a\leq X\leq b)=\int\limits^{b}_{a}f(x)dx$$
+
+for some function $f$ which satisfies $f(x)\geq0$ for all $x$ and
+
+$$\int\limits^\infty_{-\infty} f(x)dx = 1$$
+
+then $f$ is said to be a **probability density function (`p.d.f.`)** for $X$.
+
+:::
+
+:::note Definition
+
+The function
+
+$$F(x)= \int\limits^{x}_{-\infty} f(t)dt$$
+
+is the **cumulative distribution function (`c.d.f.`)**.
+
+:::
+
+### Examples
+
+:::info Example
+
+Consider a random variable $X$ from the uniform distribution, denoted by $X\sim Unf(0,1)$.
+This distribution has density:
+
+$$f(x) = \begin{cases} 1 &\text{if } 0 \leq x \leq 1\\ 0 &\text{e.w.} \end{cases}$$
+
+The cumulative distribution function is given by:
+
+$$P[X\leq x] = \int\limits^{x}_{-\infty} f(t)dt = \begin{cases} 0 & \text{if } x<0\\ x & \text{if } 0 \leq x \leq 1\\ 1 & \text{else} \end{cases}$$
+
+:::
+
+:::info Example
+
+Suppose $X \sim P(\lambda)$, where $X$ may denote the number of events per unit time.
+The `p.m.f.` of $X$ is described by
+
+$$p(x)=P[X=x]=e^{-\lambda}\frac{\lambda^x}{x!}$ for $x=0,1,2,\dots$$
+
+Let $T$ denote the waiting time between events, or simply until the first event.
+Consider the event $T>t$ for some number $t>0$.
+If $X\sim p(\lambda)$ denotes the number of events per unit time, then let $X_t$ be the number of events during the time period for $0$ through $t$.
+Then it is natural to assume $X_t \sim P(\lambda t)$ and it follows that $T>t$ if and only if $X_t=0$ and we obtain $P[T>t]=P[X_t=0]=e^{-\lambda t}$.
+It follows that the `c.d.f.` of $T$ is
+
+$$F_T(t)=P[T\leq t]=1-P[T>t]=1-e^{-\lambda t}$ for $t>0$$
+
+The `p.d.f.` of $T$ is therefore
+
+$$f_T(t)=F_T'(t)=\displaystyle\frac{d}{dt}F_T(t)=\displaystyle\frac{d}{dt}(1-e^{-\lambda t})=0-e^{- \lambda t} \cdot (-\lambda)=\lambda e^{-\lambda t}$$
+
+for $t \geq 0$ and $f_T(t)=0$ for $t < 0$
+
+The resulting density
+
+$$f(t) = \begin{cases} \lambda e^{-\lambda t} & \text{for } t \geq0\\ 0 & \text{for } t<0 \end{cases}$$
+
+describes the exponential distribution.
+
+This distribution has the expected value
+
+$$E[T]=\int_{-\infty}^{\infty} tf(t)dt=\lambda \int_{0}^{\infty} t e^{-\lambda t}dt$$
+
+Let's use integration by parts (see below), i.e.: $\int fg' = fg - \int f'g$ to solve that integral.
+Let $f=t$ and $g'=e^{-\lambda t}$.
+Then $f' = 1$ and $g=-\displaystyle\frac{e^{- \lambda t}}{\lambda}$.
+We obtain:
+
+$$\begin{aligned} &=\lambda \left( \left[ \displaystyle\frac{-te^{-\lambda t}}{\lambda}\right]_{0}^{\infty} - \int_0^\infty - \displaystyle\frac{-e^{-\lambda t}}{\lambda} dt \right)\\ &= \lambda \left( (0 - 0) - \left[ \displaystyle\frac{e^{-\lambda t}}{\lambda^2} \right]_0^\infty \right)\\ &= -\lambda \left(0 - \displaystyle\frac{1}{\lambda^2}\right)\\ &= \displaystyle\frac{1}{\lambda}\end{aligned}$$
+
+:::
+
+## Probabilities In R: The Normal Distribution
+
+R has functions to compute values of probability density functions (`p.d.f.`) and cumulative distribution functions (`c.m.d.`) for most common distributions.
+
+### Details
+
+The `p.d.f.` for the normal distribution is
+
+$$p(t)=\displaystyle\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}$$
+
+The `c.d.f.` for the normal distribution is
+
+$$\Phi(x)=\int_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt$$
+
+### Examples
+
+:::info Example
+
+`dnorm()` gives the value of the normal `p.d.f.`
+
+:::
+
+:::info Example
+
+`pnorm()` gives the value of the normal `c.d.f.`
+
+:::
+
+## Some Rules of Integration
+
+### Examples
+
+:::info Example
+
+Using integration by parts we obtain:
+
+$$\int \ln(x)x dx= \displaystyle\frac{1}{2}x^2\ln(x)-\int \frac{1}{2}x^2\cdot \displaystyle\frac{1}{x} dx = \displaystyle\frac{1}{2}x^2\ln(x)-\int \frac{1}{2}x dx=\displaystyle\frac{1}{2}x^2\ln(x)-\displaystyle\frac{1}{4}x^2$$
+
+:::
+
+:::info Example
+
+Consider $\int_1^2 2xe^{x^2} dx$.
+By setting $x=g(t)=\sqrt{t}$ we obtain
+
+$$\int_1^2 2xe^{x^2} dx = \int_1^4 2\sqrt{t}e^{t}\displaystyle\frac{1}{2\sqrt{t}}dt=\int_1^4 e^t dt=e^4-e$$
+
+:::
+
+### Handout
+
+The two most common "tricks" applied in integration are a) integration by parts and b) integration by substitution
+
+a) **Integration by parts**
+
+$$(fg)'=f'g+fg'$$
+
+by integrating both sides of the equation we obtain:
+
+$$fg=\int f'g dx + \int fg' dx \leftrightarrow \int fg' dx=fg-\int f'g dx$$
+
+b) **Integration by substitution**
+
+Consider the definite integral $\int_a^b f(x) dx$ and let $g$ be a one-to-one differential function for the interval $(c,d)$ to $(a,b)$.
+Then
+
+$$\int_a^b f(x) dx=\int_c^d f(g(y))g'(y) dy$$