Thread: Variance(U) formula if U=aX+bY+cZ+d

I don't find this formula anywhere. Wikipedia has the case where:
Var(aX+bY)=a^2*Var(X)+a^2*Var(Y)+2abCov(X,Y).
How to derive formula for my case?

Originally Posted by totalnewbie

I don't find this formula anywhere. Wikipedia has the case where:
Var(aX+bY)=a^2*Var(X)+a^2*Var(Y)+2abCov(X,Y).
How to derive formula for my case?

So you want to find

Denote



Your problem then becomes

Of course, from the formula given on wikipedia, you know how to expand:



At this point, you can further simplify the first two variance terms. But for the last one, you need to know that


Can you finish the expansion from here?