Expectation of X^Y: Estimator & Calculation

In summary, the conversation discusses the estimator for the expectation of X^Y, where X and Y are independent and identically distributed random variables with expectations E(X) and E(Y) respectively. It is mentioned that for small variances, E(X^Y) is approximately equal to E(X)^E(Y), but the accuracy of this statement is questioned. A counterexample is suggested to demonstrate that this statement may not be true.
  • #1
leehwd
2
0
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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  • #2
I'm pretty sure that's not enough information.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 

1. What is the Expectation of X^Y?

The expectation of X^Y is a statistical measure that calculates the average value of the variable X raised to the power of Y. It is also known as the expected value or mean of X^Y. This measure is commonly used in probability and statistics to understand the central tendency of a data set.

2. How is the Expectation of X^Y calculated?

The expectation of X^Y can be calculated using the formula E[X^Y] = ∑(x^y * P(x)), where x and y represent the values of X and Y respectively, and P(x) is the probability of X taking on that value. This formula is used to find the weighted average of all possible outcomes of X^Y.

3. What is the difference between an Estimator and Calculation for the Expectation of X^Y?

An estimator is a statistical method or formula used to estimate the value of the Expectation of X^Y. On the other hand, the calculation is the actual process of finding the exact value of the Expectation of X^Y using the given data or formula. The estimator provides an approximate value, while the calculation gives the exact value.

4. Why is the Expectation of X^Y important in statistics?

The Expectation of X^Y is an essential measure in statistics because it helps in understanding the average value of a variable raised to a power. It is widely used in various statistical analyses and can provide valuable insights into the central tendency of a data set. It is also used in making predictions and decisions based on probability and statistics.

5. Can the Expectation of X^Y be negative?

Yes, the Expectation of X^Y can be negative. This can happen when the values of X and Y are such that the resulting X^Y values are negative. For example, if X is a negative number and Y is an even number, then the Expectation of X^Y may result in a negative value. It is essential to consider the range of possible outcomes when calculating the Expectation of X^Y to avoid any confusion.

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