Question

1. Find the probability P(0<X₁<1/3 , 0<X₂<1/3) where X₁, X₂ have the joint pdf

                   f(x₁, x₂) = 4x₁(1-x₂) ,     0<x₁<1 0<x₂<1

                                      0,                  otherwise

(ii) For the same joint pdf, calculate E(X₁X₂) and E(X₁ + X₂)

(iii) Calculate Var(X₁X₂)

Answer 1

(ii) The expected value of X₁X₂ here is computed as:

Integrating with respect to x₁, we get here:

Now integrating with respect to x₂, we get here:

Therefore 2/9 is the expected value of X₁X₂ here.

Now the expected value of X₁ + X₂ is computed as: E(X₁) + E(X₂)

Integrating with respect to x₁, we get here:

Integrating with respect to x₂, we get here:

Therefore 1 is the expected value of X₁ + X₂ here.

(ii) The second moment of X₁X₂ here is first computed as:

Integrating with respect to x₁, we get here:

Integrating with respect to x₂, we get here:

Now the variance of X₁X₂ is computed here as:

Therefore 0.0340 is the required probability here.