# 25.3 - Sums of Chi-Square Random Variables

25.3 - Sums of Chi-Square Random VariablesWe'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 "**additive property of independent chi-squares**."

## Theorem

Let \(X_i\) denote \(n\) 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 \(r_1+r_2+\ldots+r_n\) 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 \(r_1+r_2+\ldots+r_n\) degrees of freedom. That is:

\(Y\sim \chi^2(r_1+r_2+\cdots+r_n)\)as was to be shown.

## Theorem

Let \(Z_1, Z_2, \ldots, Z_n\) have standard normal distributions, \(N(0,1)\). If these random variables are independent, then:

\(W=Z^2_1+Z^2_2+\cdots+Z^2_n\)

follows a \(\chi^2(n)\) distribution.

### Proof

Recall that if \(Z_i\sim N(0,1)\), then \(Z_i^2\sim \chi^2(1)\) for \(i=1, 2, \ldots, n\). 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, \(W\sim \chi^2(n)\), as was to be proved.

## Corollary

If \(X_1, X_2, \ldots, X_n\) are independent normal random variables with different means and variances, that is:

\(X_i \sim N(\mu_i,\sigma^2_i)\)

for \(i=1, 2, \ldots, n\). Then:

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

### 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.