Let X and Y be discrete random variables and a; b are real numbers.
Prove E(aX + bY ) = aE(X)+ bE(Y ) and Var(aX + bY ) = a^2Var(X)+
b^2Var(Y )+ 2abCov(X; Y ).
Both these seem easy but I can't proof why E(X+Y)=E(X)+E(Y)
Let X and Y be discrete random variables and a; b are real numbers.
Prove E(aX + bY ) = aE(X)+ bE(Y ) and Var(aX + bY ) = a^2Var(X)+
b^2Var(Y )+ 2abCov(X; Y ).
Both these seem easy but I can't proof why E(X+Y)=E(X)+E(Y)
Look at the definition of expectation for discrete RVs:
So (assuming suitable conditions allowing us to change orders of summation etc hold):
(here we are writing the marginal distribution and similar for the other case)