If f(x 1 , x 2 ) e x 1 x 2 , 0 x 1 , 0 x 2 , zero elsewhere, is the joint pdf of the random variables X 1 and X 2 , show that X 1 and X 2 are independent and that M(t 1 , t 2 ) (1 t 1 ) 1 (1 t 2 ) 1 , t 2 1, t 1 1 Also show that Accordingly, find the mean and the variance of Y X 1 X 2 E(et(X1 X)) (...

To show that X1 and X2 are independent we need to show that the joint pdf can be written as the prod ...

If f(x 1 , x 2 ) = e x 1 x 2 , 0 < x

Question:

If f(x₁, x₂) = e^−x₁−x_² , 0 < x₁ < ∞, 0 < x₂ < ∞, zero elsewhere, is the joint pdf of the random variables X₁ and X₂, show that X₁ and X₂ are independent and that M(t₁, t₂) = (1 − t₁)⁻¹(1 − t₂)⁻¹, t₂ < 1, t₁ < 1. Also show that