Question

Let X be chi-squared with k = 5 degrees of freedom.

(a) What is P[X > 1]? (This exact calculation requires some serious integration)

(b) Find P[X > 1] with the normal approximation, i. e., approximate that probability assuming X ~ N(mu, sigma squared), where mu and sigma squared are the mean and variance of a Gamma(5/2, 1/2).

Answer 1

a)

using R

P(X > 1) = 0.9625658

pchisq(x,df) gives P(X < x) where X follow chi-square with degree of freedom = df

 1-pchisq(1,5)
[1] 0.9625658

2)

mean = df = k = 5

variance = 2*df = 2*k = 10

P(X > 1)

= 0.8970484

 1-pnorm(1,5,sqrt(10))
[1] 0.8970484