Question

Problem 1. True/False Questions: Circle T for true and F for false.

T F If X and Y are independent standard normal random variables, then X/Y has a t(1) distribution.

T F If X and Y are independent t(1) random variables, then X + Y has the same distribution as 2X.

T F If X and Y are independent Gamma(1, 1) random variables, then X/Y has an F(2, 2) distribution.

T F If X ∼ t(8), then E(e^X) is finite. T F There are no integers m and n such that an F(m, n) random variable has a moment generating function.

T F If X ∼ t(m), then Y = −X ∼ t(m).

T F If X ∼ t(m), then 1/X² ∼ F(1, m)

T F If X ∼ F(2n, 2n), then 1/(1 + X) ∼ Beta(n, n).

T F If X ∼ F(m, n), then 1/X ∼ F(n, m).

T F If X and Y are independent F(m, n) random variables, then X + Y ∼ F(2m, 2n)

Answer 1

If X and Y are independent standard normal random variables, then X/Y has a t(1) distribution. TRUE

If X and Y are independent t(1) random variables, then X + Y has the same distribution as 2X. TRUE

If X and Y are independent Gamma(1, 1) random variables, then X/Y has an F(2, 2) distribution. FALSE

If X ∼ t(8), then E(e^X) is finite. FALSE

There are no integers m and n such that an F(m, n) random variable has a moment generating function.

If X ∼ t(m), then Y = −X ∼ t(m). FALSE

If X ∼ t(m), then 1/X² ∼ F(1, m). TRUE

If X ∼ F(2n, 2n), then 1/(1 + X) ∼ Beta(n, n). FALSE

If X ∼ F(m, n), then 1/X ∼ F(n, m). TRUE

If X and Y are independent F(m, n) random variables, then X + Y ∼ F(2m, 2n). TRUE