# Sums of Chi-Square Random Variables

We'll now turn our attention towards applying the theorem and corollary of the previous page to the case in which we have a function involving a sum of independent chi-square random variables. The following theorem is often referred to as the "**a****dditive property of independent chi-squares**."

n independent random variables that follow these chi-square distributions: *\(X_1 \sim \chi^2(r_1)\)**\(X_2 \sim \chi^2(r_2)\)*- \(\vdots\)
- \(X_n \sim \chi^2(r_n)\)
Then, the sum of the random variables: \(Y=X_1+X_2+\cdots+X_n\) follows a chi-square distribution with r_{n }degrees of freedom. That is:\(Y\sim \chi^2(r_1+r_2+\cdots+r_n)\) |

**Proof.**

We have shown that *M*_{Y}(*t*) is the moment-generating function of a chi-square random variable with *r*_{1} + *r _{2}* + ... +

*r*

_{n }degrees of freedom. That is:

\(Y\sim \chi^2(r_1+r_2+\cdots+r_n)\)

as was to be shown.

\(W=Z^2_1+Z^2_2+\cdots+Z^2_n\) follows a χ |

**Proof. **Recall that if *Z _{i }*~

*N*(0, 1), then

*Z*

_{i}^{2}~ χ

^{2}(1) for

*i*= 1, 2, ... ,

*n*. Then, by the additive property of independent chi-squares:

\(W=Z^2_1+Z^2_2+\cdots+Z^2_n \sim \chi^2(1+1+\cdots+1)=\chi^2(n)\)

That is, *W* ~ χ^{2}(*n*), as was to be proved.

\(X_i \sim N(\mu_i,\sigma^2_i)\) for \(W=\sum\limits_{i=1}^n \dfrac{(X_i-\mu_i)^2}{\sigma^2_i} \sim \chi^2(n)\) |

**Proof. **Recall that:

\(Z_i=\dfrac{(X_i-\mu_i)}{\sigma_i} \sim N(0,1)\)

Therefore:

\(W=\sum\limits_{i=1}^n Z^2_i=\sum\limits_{i=1}^n \dfrac{(X_i-\mu_i)^2}{\sigma^2_i} \sim \chi^2(n)\)

as was to be proved.