Question

Assume that a random variable. X, follow a chi-square distribution with 23 degrees of freedom. Then

use this information to determine the following:

(a) P(14.85 < X<32.01).

(b) Constants and b such that Pa< X<b) = 0.95 and PX a) = 0.025. (c) The mean and variance of X.

Answer 1

(a) $\nu=23$

$$P(14.85<X<32.01)=P(X<32.01)$$

$$-P(X\leq14.85)=0.900064-0.100071$$

$$=0.799993$$

(b) $\nu=23$

$$P(X<a)=0.025$$

$$a=18.1373$$

$$P(a<X<b)=P(X<b)-P(X\leq a)$$

$$=P(X<b)-0.025=0.95$$

$$P(X<b)=0.975$$

$$b=38.075$$

(c) $\nu=23$

The mean and variance of the chi-squared distribution are

$$\mu=\nu=23$$

$$Var(X)=\sigma^2=23^2=529$$