# Chi-Square Distribution

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