# Chi-Square Distribution

Suppose we have n independently and identically distributed standard normal random variables ${\displaystyle X_{1},\dots ,X_{n}:X_{i}\sim N(0;1)}$ for ${\displaystyle i=1,\dots ,n}$. where n is a positive integer. The distribution of the sum of the squared ${\displaystyle X_{i}s}$ ${\displaystyle Y=X_{1}^{2}+X_{2}^{2}+\dots X_{n}^{2}}$ is referred to as the ${\displaystyle \chi ^{2}}$ Chi-square distribution, with parameter df, or written shortly as ${\displaystyle \chi ^{2}(df)}$. The parameter ${\displaystyle df}$ represents the degrees of freedom of the distribution, with ${\displaystyle df>0}$. The expected value and variance of Chi-square distribution are given as ${\displaystyle E(Y)=df{\text{ a }}Var(Y)=2df.}$ This diagram plots the density functions for some Chi-square distributions, with different values for the degrees of freedom ${\displaystyle df}$.
The Chi-square, t-, and F- distributions are distributions that are functions of Normal random variables that are particularly useful in statistics. On the chi-square distribution. The parameter ${\displaystyle df}$ denotes the degrees of freedom. The degrees of freedom reflects the number of independent random variables included in the sum Y. If the random variables ${\displaystyle X_{i},i=1,\ldots ,n}$ are independent from each other, then squaring and summing them does not change their properties. In this example, the random variable ${\displaystyle Y}$: ${\displaystyle Y=X_{1}^{2}+X_{2}^{2}+\dots +X_{n}^{2}}$ will have the Chi-square distribution with ${\displaystyle df=n}$ degrees of freedom. The shape of the density function will depend on the parameter ${\displaystyle df}$. For ${\displaystyle df=1}$ and ${\displaystyle df=2}$, the ${\displaystyle \chi ^{2}}$distribution follows a monoton structures. For small values of ${\displaystyle df}$, the ${\displaystyle \chi ^{2}}$ -distribution will be skewed to the right. However, as ${\displaystyle df}$ increases the ${\displaystyle \chi ^{2}}$-distribution will tend towards the normal density function. The ${\displaystyle \chi ^{2}}$-distribution is tabulated for a number of values of ${\displaystyle df}$.