Please show working out with english in complete detail Than

Please show working out with english in complete detail, Thank you :)

(1) Let X₁, X₂, and X₃ be three independant and identically distributed random variables with

We form a new sequaence of random variables

Y₁ = X₁ , Y₂ = X₁X₂ and Y₃ = X₁X₂X₃.

(a) Obtain the marginal PMFs of Y₂ and Y₃.

(b) Obtain the conditional PMFs of Y₂ and Y₃ given Y₁ = y₁ for y₁ = -1,1.

(c) Obtain the joint PMFs of each of (Y₁,Y₂), (Y₁,Y₃) and (Y₂,Y₃).

(d) Compute E[Y₁Y₂], E[Y₁Y₃] and E[Y₂Y₃].

(e) Are Y₁ and Y₂ independant? Are Y₁ and Y₃ independant? Are Y₂ and Y₃ independant?

Solution