Add content for multivariate-to-power/distributions-normal

Add actual content to the skeleton of the
`multivariate-to-power/distributions-normal` section.

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

chapters/multivariate-to-power/distributions-normal/reading/read.md

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+# Some Distributions Related to the Normal
+
+## The Normal and Sums of Normals
+
+The sum of independent normally distributed random variables is also normally distributed.
+
+### Details
+
+The sum of independent normally distributed random variables is also normally distributed.
+More specifically, if $X_1 \sim N(\mu_1, \sigma_{1}^2)$ and $X_2 \sim N(\mu_2, \sigma_{2}^2)$ are independent, then $X_1 + X_2 \sim N(\mu, \sigma^2)$, since $\mu = E \left[ X_1 + X_2 \right] = \mu_1 + \mu_2$ and\ $\sigma^2 = Var \left[ X_1 + X_2 \right]$ with $\sigma^2 = \sigma_{1}^2 + \sigma_{2}^2$ \ if $X_1$ and $X_2$ are independent.
+
+Similarly
+
+$$\displaystyle\sum_{i=1}^{n} X_i$$
+
+is normal if $X_1, \ldots, X_n$ are normal and independent.
+
+### Examples
+
+:::info Example: Simulating and plotting a single normal distribution.
+
+$Y \sim N(0,1)$
+
+```text
+library(MASS) # for truehist
+par(mfcol=c(2,2))
+y <- rnorm(1000) # generating 1000 N(0,1)
+mn <- mean(y)
+vr <- var(y)
+truehist(y,ymax=0.5) # plot the histogram
+xvec <-seq(-4,4,0.01) # generate the x-axis
+yvec <- dnorm(xvec) # theoretical N(0,1) density
+lines(xvec,yvec,lwd=2,col="red")
+ttl <- paste("Simulation and theory N(0,1)\n", "mean=",round(mn,2), "and variance=",round(vr,2))
+title(ttl)
+```
+
+:::
+
+:::info Example: Sum of two normal distributions
+
+$$Y_1 \sim N(2, 2^2)$$
+
+and
+
+$$Y_2 \sim N(3, 3^2)$$
+
+```text
+y1 <- rnorm(10000,2,2) # N(2,2^2)
+y2 <- rnorm(10000,3,3) # N(3, 3^2)
+y <- y1+y2
+truehist(y)
+xvec <- seq(-10,20,0.01)
+mn<-mean(y)
+vr <- var(y)
+cat("The mean is",mn,"\n")
+cat("The variance is ",vr,"\n")
+cat("The standard deviation is", sd(y), "\n")
+yvec <- dnorm(xvec,mean=5,sd=sqrt(13)) # N() density
+lines(xvec,yvec,lwd=2,col="red")
+ttl <- paste("The sum of N(2,2^2) and N(3,3^2)\n", "mean=",round(mn,2), "and variance=", round(vr,2))
+title(ttl)
+```
+
+:::
+
+:::info Example Sum of nine normal distributions, all with $\mu = 42$ and $\sigma^2=2^2$.
+
+```text
+ymat <- matrix(rnorm(10000*9,42,2),ncol=9)
+y <- apply(ymat,1,mean)
+truehist(y)
+mn <- mean(y)
+vr <- var(y)
+cat("The mean is",mn,"\n")
+cat("The variance is ",vr,"\n")
+cat("The standard deviation is",sd(y),"\n")
+# plot the theoretical curve
+xvec <- seq(39,45,0.01)
+yvec <- dnorm(xvec,mean=5,sd=sqrt(13)) # N() density
+lines(xvec,yvec,lwd=2,col="red")
+ttl <- paste("The sum of nine N(42^2) \n", "mean=",round(mn,2), "and variance=",round(vr,2))
+title(ttl)
+```
+
+:::
+
+## The Chi-square Distribution
+
+If $X \sim N(0,1)$,then $Y = X^2$ has a distribution which is called the chi-square distribution ( $\chi^2$ ) on one degree of freedom.
+This can be written as:
+
+$$Y \sim \chi^2$$
+
+![Fig. 34](../media/21_2_The_Chi-square_distribution.png)
+
+### Details
+
+:::note Definition
+
+If $X_1, X_2, \ldots, X_n$ are i.i.d. $N(0,1)$ then the distribution of $Y = X_1^2 + X_1^2 + \ldots + X_n^2$ has a **square ( $\chi^2$ )distribution**.
+
+:::
+
+## Sum of Chi-square Distributions
+
+Let $Y_1$ and $Y_2$ be independent variables.
+If $Y_1 = \chi^2_{\nu_1}$ and $Y_2 = \chi^2_{\nu_2}$, then the sum of these two variables also follows a chi-squared ( $\chi^2$) distribution:
+
+$$Y_1 + Y_2 = \chi^2_{\nu_1+ \nu_2}$$
+
+![Fig. 35](../media/21_3_Sum_of_Chi_square_Distributions.png)
+
+### Details
+
+:::note Note
+
+Recall that if
+
+$$X_1, \ldots, X_n \sim N (\mu, \sigma^2)$$
+
+are i.i.d., then
+
+$$\displaystyle\sum_{i=1}^n \left ( \displaystyle\frac {\bar{X} - \mu} {\sigma}\right ) ^2= \displaystyle\sum_{i=1}^n \displaystyle\frac {\left ( \bar{X} - \mu\right ) ^2} {\sigma}\sim \chi^2$$
+
+:::
+
+## Sum of Squared Deviation
+
+If $X_1,\cdots,X_n \sim N(\mu,\sigma^2)$ i.i.d, then
+
+$$\displaystyle\sum_{i=1}^n \left ( \displaystyle\frac{X_i-\mu}{\sigma} \right )^2 \sim \chi_{n}^2,$$
+
+but we are often interested in
+
+$$\displaystyle\frac{1}{n-1}\displaystyle\sum_{i=1}^n (X_i-\bar{X})^2\sim \chi_{n-1}^2$$
+
+### Details
+
+Consider a random sample of Gaussian random variables, i.e. $X_1,\cdots,X_n \sim N(\mu,\sigma^2)$ i.i.d.
+Such a collection of random variables have properties which can be used in a number of ways.
+
+$$\displaystyle\sum_{i=1}^n \left ( \displaystyle\frac{X_i-\mu}{\sigma} \right )^2 \sim \chi_{n}^2$$
+
+but we are often interested in
+
+$$\displaystyle\frac{1}{n-1}\displaystyle\sum_{i=1}^n (X_i-\bar{X})^2\sim \chi_{n-1}^2$$
+
+:::note Note
+
+A degree of freedom is lost because of subtracting the estimator of the mean as opposed to the true mean.
+
+:::
+
+The correct notation is:
+
+$\mu$ := population mean
+
+$\bar{X}$ := sample mean (a random variable)
+
+$\bar{x}$ := sample mean (a number)
+
+## The $T$ distribution
+
+If $U\sim N(0,1)$ and $W\sim\chi^{2}_{\nu}$ are independent, then the random variable
+
+$$T=\displaystyle\frac{U}{\sqrt{\displaystyle\frac{w}{\nu}}}$$
+
+has a distribution which we call the $T$ distribution on $\nu$ degrees of freedom denoted $T \sim t_{\nu}$.
+
+### Details
+
+:::note Definition
+
+If $U\sim N(0,1)$ and $W\sim\chi^{2}_{\nu}$ are independent, then the random variable
+
+$$T:=\displaystyle\frac{U}{\sqrt{\displaystyle\frac{w}{\nu}}}$$
+
+has a distribution which we call the $T$ distribution on $\nu$ degrees of freedom, denoted $T \sim t_\nu$.
+
+:::
+
+It turns out that if $X_1, \ldots,X_n \sim N(\mu,\sigma ^2)$ and we set:
+
+$$\bar{X}=\displaystyle\frac{1}{n}\displaystyle\sum_{i=1}^n X_i$$
+
+and
+
+$$S= \sqrt{\displaystyle\frac{1}{1-n}\displaystyle\sum_{i=1}^n (X_i-X)^2}$$
+
+then
+
+$$\displaystyle\frac{\bar{X}-\mu}{S/\sqrt{n}} \sim t_{n-1}$$
+
+This follows from $\bar{X}$ and $\Sigma_{i=1}^N(X_i-\bar{X})^2$ being independent and $\displaystyle\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1)$, $\displaystyle\sum \displaystyle\frac{(X_i-\bar{X})^2}{\sigma^2}\sim \chi_{n-1}^2$.