# Normal Distribution Theory

### Sum of normal variables

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

Then if

${\displaystyle Y=\sum _{i=1}^{n}{a_{i}}{x_{i}}+b}$

we have the distribution:

${\displaystyle Y\sim N(\sum _{i=1}^{n}{a_{i}}{\mu _{i}}+b;\sum _{i=1}^{n}{{a_{i}}^{2}}{{\sigma _{i}}^{2}})}$

### Chi-squared distribution

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

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

${\displaystyle Z\sim \chi ^{2}(n)}$

### t-distribution

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

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

${\displaystyle Z\sim {t(n)}}$

### F-distribution

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

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

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