Maybe I've answered my own question ... I guess I can't say the Xi are independent of themselves :-) So, my first approach was invalid.
I am working on a question that has me stumped. X1, X2, ... Xn are normally distributed and independent. Each has a mean of 0 and variance sigma^2. I am asked to find E(Xi^2) and V(Xi^2). There seem to me two ways to approach this:
First, since the Xi are independent, we could say E(Xi^2) = E(Xi)*E(Xi) = 0*0 = 0. Second, we could also say V(Xi) = E(Xi^2) - [E(Xi)]^2 = E(Xi^2) - 0 = E(Xi^2) in which case E(Xi^2) = V(Xi) = sigma^2.
From these two, we are led to conclude that sigma must be 0 (in which case this is a uniform distribution, not normal). Have I made some error in thinking or is this problem flawed?
Maybe I've answered my own question ... I guess I can't say the Xi are independent of themselves :-) So, my first approach was invalid.
I knew I would need that formula. The only problem is that I have no idea how to find E(Xi^4) given that I know E(Xi^2).
There are lots of tricks to get what you want. But if you know E(X^2) then it's possible to get E(X^4). This might help.
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