The random variable Q is uniform on [0, 1]. Conditioned on Q = q, the random variable X is Bernoulli with parameter q.

Question

a) The conditional variance, Var(X | Q), is equal to:

b) Recall that a uniform random variable on [0, 1] has a variance of 1/12 and also satisfies E[Q^2] = 1/3. Then:

Var(E [X | Q]) =
E[Var(X | Q)] =

Answer 1

The random variable X has a PDF of the form:

fX(x) = {1/x^2, for x >= 1,
0, otherwise.}

Let Y = X^2. For Y >= 1, the PDF of Y takes the form fY(y) = a/y^b. Find the values of a and b.

a =
b =