Let XiN01 i123 be 3 iid standard normal random variables a W

Let X_i~N(0,1), i=1,2,3 be 3 iid standard normal random variables.

a. What is the distribution of Q₁=X₁²?

b. What is the distribution of  Q2=X₂²?+X₃²?

c. What is the distribution of Q₃= X₁²+X₂²+X₃²?

d. What is the distribution of G=2Q₁/Q₂?

e. Explain why (Q₃/3)/(Q₂/2) is NOT an F-distributed random variable?

Thank you so much for your help!

Solution

a) Distribution of Q1 is the distribution of square of a std normal variate

Hence chi square distribution.

b) Sum of squares of 2 std normal variates. Hence again chi square distribution.

c) Sum of squares of 3 std normal variates. Hence again chi square distribution.

d) 2Q1/Q2 would be ratio of chi square distribution.

So it is F distribution.

e)