Question

1. (a) Which of the following is true and which is false? If you think a statement is true, provide a proof. Disprove those you think are false by giving a counterexample (i) A probability density function never exceeds 1 (ii) Suppose X and Y are two random variables defined on the same sample space, such that X > Y. This implies Var[X > Var Y] (ii) Let Z be a standard normal random variate N(0, 1). Then Z and -Z have the same distribution

Answer 1

a) false

let us say X= uniform ( 0,0.5)

then pdf = 1/ (0.5 -0) = 2   > 1

b)

false

X = {5,6} with p = 1/2

Y = {1,2,3,4} with p =1/4

hence var(X) < Var(Y)

c)

true