For random variables X and Y does VX Y always equal VX VY

For random variables X and Y, does V(X + Y) always equal V(X) + V(Y)? What about E(X + Y) and E(X) + E(Y)?

- For random variables X and Y, does E(XY) always equal E(X)E(Y)? How is this related to V(X + Y)?

- Is it true for any function f that E(f(X)) = f(E(X))? For what kind of functions do we know this is true?

- For constants a and b, does V(aX + b)= aV(X) + b? How about for expected value?

- If we have a random variable X that can only take the values x = 0, 1, 2, 3, or 4, then what is P(X=5)? What is P(X?4)?

Solution

when X and Y are independent, only then V(X+Y)=V(X)+V(Y)

E(X+Y)=E(X)+E(Y) (always)

when X and Y are independent, only then E(XY) = E(X)E(Y)

V(aX + b)= a^2V(X)

P(X=5)=0