Expectation of X^Y: Estimator & Calculation

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SUMMARY

The expectation of the expression E(X^Y), where X and Y are independent and identically distributed (iid) random variables, is a complex topic. The discussion reveals that while the Taylor expansion suggests E(X^Y) = E(X)^E(Y) for small variances, this is not universally valid. A counterexample using uniformly distributed variables indicates that the assumption can lead to nonsensical results, particularly when considering distributions that are symmetrical around zero. Therefore, the relationship between E(X^Y) and E(X)^E(Y) requires careful consideration of the distributions involved.

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  • Understanding of independent and identically distributed (iid) random variables
  • Familiarity with expectation notation and properties
  • Knowledge of Taylor expansion in probability theory
  • Basic concepts of probability distributions, particularly uniform distributions
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  • Explore the properties of expectation for non-linear transformations of random variables
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  • Investigate specific cases of E(X^Y) for various distributions, including normal and uniform
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Mathematicians, statisticians, and data scientists interested in advanced probability theory and the behavior of expectations in complex random variable scenarios.

leehwd
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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.
 
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I'm pretty sure that's not enough information.
 
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.
 
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.
 
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(XY).
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.
 

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