# Normal Distribution Theory

### Sum of normal variables

Suppose that $x_{i}\sim N(\mu _{i};{\sigma _{n}}^{2})$ i=1,...,n and suppose that the n random variables are independent.

Then if

$Y=\sum _{i=1}^{n}{a_{i}}{x_{i}}+b$ we have the distribution:

$Y\sim N(\sum _{i=1}^{n}{a_{i}}{\mu _{i}}+b;\sum _{i=1}^{n}{{a_{i}}^{2}}{{\sigma _{i}}^{2}})$ ### Chi-squared distribution

Let $x_{1},x_{2},...,x_{n}$ be independent and identically distributed $\sim N(0;1)$ and consider

$Z={x_{1}}^{2}+{x_{2}}^{2}+...+{x_{n}}^{2}$ The distribution of $Z$ is called Chi-squared distribution with n degrees of freedom.

$Z\sim \chi ^{2}(n)$ ### t-distribution

Let $x$ and $x_{1},x_{2},...,x_{n}$ be independent and identically distributed $\sim N(0;1)$ and consider

$Z={{x} \over {\sqrt {({x_{1}}^{2}+{x_{2}}^{2}+...+{x_{n}}^{2})/n}}}$ The distribution of $Z$ is called t-distribution with n degrees of freedom.

$Z\sim {t(n)}$ ### F-distribution

Let $x_{1},x_{2},...,x_{n}$ and $y_{1},y_{2},...,y_{n}$ be independent and identically distributed $\sim N(0;1)$ and consider

$Z={({x_{1}}^{2}+{x_{2}}^{2}+...+{x_{n}}^{2})/n \over ({y_{1}}^{2}+{y_{2}}^{2}+...+{y_{m}}^{2})/m}$ The distribution of $Z$ is called F-distribution with n and m degrees of freedom.

$Z\sim {F(n,m)}$ 