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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 Section

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))
1.3 Applications Of Square Rootsmr. Mac

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:

(Ysim 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:

(Ysim chi^2(r_1+r_2+cdots+r_n))

as was to be shown.

Theorem Section

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_isim N(0,1)), then (Z_i^2sim 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, (Wsim chi^2(n)), as was to be proved.

Corollary Section

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=sumlimits_{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=sumlimits_{i=1}^n Z^2_i=sumlimits_{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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