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

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

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

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

@@ -1 +1,159 @@
 # 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
+
+$$\sum_{i=1}^{n} X_i$$
+
+is normal if $X_1, \ldots, X_n$ are normal and independent.
+
+### Examples
+
+:::info Example: Simulating and plotting a single normal distribution.
+
+$Y \sim N(0,1)$
+
+```
+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
+
+$$Y_2 \sim N(3, 3^2)$$
+
+```
+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) # check 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 $\sigma^2=2^2$.
+
+```
+ymat<-matrix(rnorm(10000*9,42,2),ncol=9) y<-apply(ymat,1,mean) truehist(y) # check 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
+
+$$\sum_{i=1}^n \left ( \frac {\bar{X} - \mu} {\sigma}\right ) ^2= \sum_{i=1}^n \frac {\left ( \bar{X} - \mu\right ) ^2} {\sigma}\sim \chi^2$$
+
+:::
+
+## Sum of Squared Deviation
+
+If $X_1,\cdots,X_n \sim N(\mu,\sigma^2)$ i.i.d, then
+
+$$\sum_{i=1}^n \left ( \frac{X_i-\mu}{\sigma} \right )^2 \sim \chi_{n}^2,$$
+
+but we are often interested in
+
+$$\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2\sim \chi_{n-1}^2$$
+
+### 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.
+
+$$\sum_{i=1}^n \left ( \frac{X_i-\mu}{\sigma} \right )^2 \sim \chi_{n}^2$$
+
+but we are often interested in
+
+$$\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2\sim \chi_{n-1}^2.$$
+
+:::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
+
+$$T=\frac{U}{\sqrt{\frac{w}{\nu}}}$$
+
+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
+
+$$T:=\frac{U}{\sqrt{\frac{w}{\nu}}}$$
+
+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:
+
+$$\bar{X}=\frac{1}{n}\sum_{i=1}^n X_i$$
+
+and
+
+$$S= \sqrt{\frac{1}{1-n}\sum_{i=1}^n (X_i-X)^2}$$
+
+then
+
+$$\frac{\bar{X}-\mu}{S/\sqrt{n}} \sim t_{n-1}$$
+
+This follows from $\bar{X}$ and $\Sigma_{i=1}^N(X_i-\bar{X})^2$ being independent and $\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1)$, $\sum \frac{(X_i-\bar{X})^2}{\sigma^2}\sim \chi_{n-1}^2$.