25.3 - Sums of Chi-Square Random Variables

25.3 - 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 "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.

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