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+# The Central Limit Theorem and Related Topics
+
+## The Central Limit Theorem
+
+If measurements are obtained independently and come from a process with finite variance, then the distribution of their mean tends towards a Gaussian (normal) distribution as the sample size increases.
+
+![Fig. 30](../media/18_1_The_Central_Limit_Theorem.png)
+
+Figure: The standard normal density
+
+### Details
+
+The **Central Limit Theorem** states that if $X_1, X_2, \ldots$ are independent and identically distributed random variables with mean $\mu$ and (finite) variance $\sigma^2$, then the distribution of $\bar{X}_n:= \frac{X_1+\dots+X_n}{n}$ tends towards a normal distribution.
+It follows that for a large enough sample size $n$, the distribution random variable $\bar{X}_n$ can be approximated by $N(\mu,\sigma^2/n)$.
+
+The standard normal distribution is given by the p.d.f.:
+
+$$\varphi(z) = \frac{1}{\sqrt{2\pi}} e^{\frac{-z^2}{2}}$$
+
+for $z\in \mathbb{R}$
+
+The standard normal distribution has an expected value of zero:
+
+$$\mu = \int z\varphi (z)dz =0$$
+
+and a variance of:
+
+$$\sigma^2 = \int ({z-\mu})^2 \varphi(z)dz=1$$
+
+If a random variable $Z$ has the standard normal (or Gaussian) distribution, we write $Z\sim N(0,1)$.
+
+If we define a new random variable, $Y$, by writing $Y=\sigma Z + \mu$, then $Y$ has an expected value of $\mu$, a variance of $\sigma^2$ and a density (p.d.f.) given by the formula:
+
+$$f(y) = \frac{1}{\sqrt{2\pi}\sigma} \ e^{\frac{-(y-\mu)^2}{2\sigma^2}}.$$
+
+This is general normal (or Gaussian) density, with mean $\mu$ and variance $\sigma^2$.
+
+The Central Limit Theorem states that if you take the mean of several independent random variables, the distribution of that mean will look more and more like a Gaussian distribution (if the variance of the original random variables is finite).
+
+More precisely, the cumulative distribution function of:
+
+$$\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}}$$
+
+converges to $\Phi$, the $N(0,1)$ cumulative distribution function.
+
+### Examples
+
+:::info Example
+
+If we collect measurements on waiting times, these are typically assumed to come from an exponential distribution with density
+
+$$f(t)=\lambda e^{-\lambda t},\textrm{ for } t>0$$
+
+The Central Limit Theorem states that the mean of several such waiting times will tend to have a normal distribution.
+
+:::
+
+:::info Example
+
+We are often interested in computing:
+
+$$w=\frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}$$
+
+which comes from a $T$ distribution (see below), if the $x_i$ are independent outcomes from a normal distribution.
+
+However, if $n$ is large and $\sigma^2$ is finite then $w$ values will look as though they came from a normal distribution.
+This is in part a consequence of the Central Limit Theorem, but also of the fact that $s$ will become close to $\sigma$ as $n$ increases.
+
+:::
+
+## Properties of the Binomial and Poisson Distributions
+
+The binomial distribution is really a sum of $0$ and $1$ values (counts of failures $= 0$ and successes $=1$).
+So, a simple, single binomial outcome will correspond to coming from a normal distribution if the count is large enough.
+
+### Details
+
+Consider the binomial probabilities:
+
+$$p(x)=\binom{n}{x}p^x(1-p)^{n-x}$$
+
+for $x=0,1,2,3, \cdots,n$ where $n$ is a non-negative integer.
+Suppose $p$ is a small positive number, specifically consider a sequence of decreasing $p$ -values, specified with $p_n= \frac{\lambda}{n}$ and consider the behavior of the probability as $n \rightarrow \infty$.
+We obtain:
+
+$$\begin{aligned} \binom{n}{x}p_n^x(1-p_n)^{n-x}& = &\frac{n!}{x!(n-x!)} \left ( \frac{\lambda}{n} \right )^x \left ( 1-\frac{\lambda}{n} \right )^{n-x}\\ & = &\frac{N(n-1)(n-2)\cdots (n-x+1)}{x!} \frac{\frac{\lambda}{n}^x}{\left ( 1-\frac{\lambda}{n} \right ) ^x} \left ( 1-\frac{\lambda}{n} \right )^n\\ & = &\frac{N(n-1)(n-2)\cdots (n-x+1)}{x!n^x} \frac{\lambda^x}{\left ( 1-\frac{\lambda}{n} \right ) ^x} \left ( 1-\frac{\lambda}{n} \right )^n\end{aligned}$$
+
+:::note Note
+
+Notice that $\frac{N(n-1)(n-2)\cdots (n-x+1)}{n^x}\to 1$ as $n\to\infty$.
+Also notice that $(1-\frac{\lambda}{n})^x\to 1$ as $n\to\infty$.
+Also
+
+$$\lim_{n \to \infty} \bigg( 1-\frac{\lambda}{n} \bigg) = e^{- \lambda}$$
+
+and it follows that
+
+$$\lim_{n \to \infty} \binom{n}{x}p_n^x(1-p_n)^{n-x} = \frac{e^{- \lambda} \lambda^x}{x!}, x= 0,1,2, \cdots, n$$
+
+and hence the binomial probabilities may be approximated with the corresponding Poisson.
+
+:::
+
+### Examples
+
+:::info Example
+
+The mean of a binomial `(n,p)` variable is $\mu=n\cdot p$ and the variance is $\sigma^2=np(1-p)$.
+
+The R command `dbinom(q,n,p)` calculates the probability of `q` successes in `n` trials assuming that the probability of a success is `p` in each trial (binomial distribution), and the R command `pbinom(q,n,p)` calculates the probability of obtaining `q` or fewer successes in `n` trials.
+
+The normal approximation of this distribution can be calculated with `pnorm(q,mu,sigma)` which becomes `pnorm(q,np,sqrt(np(1-p))`.
+Three numerical examples (note that pbinom and pnorm give similar values for large n):
+
+```
+pbinom(3,10,0.2) [1] 0.8791261 pnorm(3,10*0.2,sqrt(10*0.2*(1-0.2))) [1] 0.7854023
+```
+
+```
+pbinom(3,20,0.2) [1] 0.4114489 pnorm(3,20*0.2,sqrt(20*0.2*(1-0.2))) [1] 0.2880751
+```
+
+```
+pbinom(30,200,0.2) [1] 0.04302156 pnorm(30,200*0.2,sqrt(200*0.2*(1-0.2))) [1] 0.03854994
+```
+
+:::
+
+:::info Example
+
+We are often interested in computing $w=\frac{\bar{x}-\mu}{s/\sqrt{n}}$, which has a $T$ distribution if the $x_i$ are independent outcomes from a normal distribution.
+If $n$ is large and $\sigma^2$ is finite, this will look as if it comes from a normal distribution.
+
+The numerical examples below demonstrate how the $T$ distribution approaches the normal distribution.
+
+```
+qnorm(0.7) [1] 0.5244005 #This is the value which gives the cumulative probability of p=0.7 for a n~(0,1) qt(0.7,2) [1] 0.6172134 #The value, which gives the cumulative probability of p=0.7 with n=2 for the t distribution. qt(0.7,5) [1] 0.5594296 qt(0.7,10) [1] 0.541528 qt(0.7,20) [1] 0.5328628
+```
+
+```
+qt(0.7,100) [1] 0.5260763
+```
+
+:::
+
+## Monte Carlo Simulation
+
+If we know an underlying process we can simulate data from the process and evaluate the distribution of any quantity based on such data.
+
+![Fig. 31](../media/18_3_Monte_Carlo_simulation.png)
+
+Figure: A simulated set of $t$-values based on data from an exponential distribution.
+
+### Examples
+
+:::info Example
+
+Suppose our measurements come from an exponential distribution and we want to compute
+
+$$t = \frac{\overline x - \mu}{s / \sqrt{n}}$$
+
+but we want to know the distribution of those when $\mu$ is the true mean.
+
+For instance, $n=5$ and $\mu = 1$, we can simulate (repeatedly) $x_1, \ldots, x_5$ and compute a $t$ -value for each.
+The following R commands can be used for this:
+
+```
+library(MASS) n<-5 mu<-1 lambda<-1 tvec<-NULL for(sim in 1:10000){ x<-rexp(n,lambda) xbar<-mean(x) s<-sd(x) t<-(xbar-mu)/(s/sqrt(n)) tvec<-c(tvec,t) }
+```
+
+```
+truehist(tvec) #truehist gives a better histogram sort(tvec)[9750] sort(tvec)[250]
+```
+
+:::