Estimator for E(X^Y) with iid random variables

[leehwd]

Can anyone let me know the estimator for the expectaiion of X^Y. X and Y are iid random variables, and their expectation are E(X) and E(Y) respectively.

Thank you.

 

[Hurkyl]

I'm pretty sure that's not enough information.

 

[leehwd]

Okay, let me put this way. Let E(X) be the expectation of random variable X, and X and Y are independent and identically distributed random variables. My question is, what is E(X^Y)? I did Talyor expansion of X^Y and concluded that, for small variances for X and Y, E(X^Y)=E(X)^E(Y).

Is this correct?
Thank you for your help in advance.

 

[Hurkyl]

It's (probably) true that if the distributions of X and Y are "close" to constant, then then E(X^Y)=E(X)^E(Y) is approximately true.

My gut says that small variances isn't enough, but I haven't done the calculations to be sure.

 

[mathman]

I believe it is not true. Try a simple particular case. For example assume X and Y are uniformly distributed over some interval, and work out E(X^Y).
A good crazy example would use two intervals symmetrical around 0 (avoid 0 itself), then E(X)^E(Y) would be 0⁰, which would be nonsense.