Exercise 3.2 Let X, Y, and Z be independent random variables with the probabilit

Question

Exercise 3.2 Let X, Y, and Z be independent random variables with the probability densities fx, fr, and fz, respectively. Let the random variable W be defined as the sum of the random variables X and Y: Show that W and Z are independent random variables. o The restriction in Exercise 3.2 to random variables which have probability densities was done purely for the sake of manipulative convenience; the final result holds just as well for the case of independent random variables with general probability distribution functions.

Explanation / Answer

Given X , Y and Z are independent

Hence ,

cov(X,Z) = E(XZ) - E(X)E(Z) = 0 => E(XZ) = E(X)E(Z) ---- (1)

and  

cov(Y,Z) = E(YZ) - E(Y)E(Z) = 0 => E(YZ) = E(Y)E(Z) ------ (2)

Now ,

cov(W,Z) = E(WZ) - E(W)E(Z)

= E((X+Y)(Z)) - E(X+Y)E(Z)

= E(XZ +YZ) - (E(X) +E(Y)) E(Z)

= E(XZ) + E(YZ) -  (E(X) E(Z) +E(Y) E(Z))

= (E(X)E(Z) + E(Y) E(Z)) - (E(X) E(Z) +E(Y) E(Z)) (from 1 and 2 )

= 0

Hence W and Z are independent.

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