Thread: stochastic independence and random variables

1. stochastic independence and random variables

Suppose X1 and X2 are stoch. indep. Bernoulli RVs with params p1 and p2 respectively. Let Y = X1X2 and W = X1 + X2. I need to determine whether the following statements are true or false so just give me a clue please, I'd like to work it out myself. If no tip can be given without giving it away please say so.
1. the RV Y ~ Bernoulli(p_1p_2)

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2. What is the general formula for finding the variance of a RV of the form Z = XY (X and Y both RVs).

2. here's what I got for my first question:
$\displaystyle E(Y) = E(X_1X_2) = E(X_1)E(X_2) = p_1p_2$
=> true, since E(X) = p for RV ~ Bernoulli(p)

3. I found the formula for the second question.

The problem wanted me to prove that given X,Y indep. and Z=XY
$\displaystyle V(Z)=E^2(X)V(Y) + E^2(Y)V(X)+V(X)V(Y)$

and the formula I found was
$\displaystyle V(XY)=E^2(X)V(Y) + E^2(Y)V(X)+V(X)V(Y)$
$\displaystyle +2E(X)E(Y)Cov(X,Y)+Cov^2(X,Y)$

and from here the answer is obvious given that Cov(X,Y)=0.

However, I'd like to know how the formula for V(XY) is derived.