Let $X, Y, Z$ be three stochastically independent random variables that are quadratic integrable (quadratintegriertbar is the German term, I didn't find a exact translation). No which statements hold always true?
- (1) $XY$ and $Z$ are stochastically independent
- (2) $XY$ and $XZ$ are stochastically independent
- (3) $Var(XZ) = Var(X)Var(Y)$
- (4) $E(XZ) = E(X)E(Z)$
- (5) $E(\frac{X}{Y^2+1}) = \frac{E(X)}{E(Y^2+1)}$
My weak guess is that statement 1 and 2 and 4 are correct, but either it is written in my uni script (There's a proof for statement 4) or I found some forumla in the internet.
But to begin, I am not quite sure how to grasp the concept of a product of independent random variables. I found this wiki article, subsection Derivation, but the formula there doens't explain anything to me. How can I see that $f_{Z}(z)$ is also stochstically independent? Surely, the equation $P(XY, Z) = P(XY)P(Z)$ must hold, but how to show it using the above equation?
Can anyone try to explain me the simple algebraic operations of random variables, such that the formulas make sense?
Anything that is intuitively accessible would be extremely appreciated.
Thanks in advance