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Mathematical Expectation

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I am having trouble with some estimators. I have defined a random variable Z such that Z=Y/X, where X and Y are random variables. I know that E(Y given X)=theta*X, where theta is an unknown parameter. I have worked out that E(Z)=theta.

Now I am having trouble with an estimator defined as W=Ybar/Xbar, where Ybar and Xbar refer to the sample averages of Y and X, respectively.

I am trying to show that E[W]=theta, and I am stuck because the expected value of a ratio is not the ratio of the expected values.

I would be very grateful if you could indicate how you would do it.

ADDENDUM

I was asked whether I was certain that E[W]=theta. This was suggested to me, but it may not be true. If anyone feels able to prove that W is biased in theta, then I would be grateful to receive the proof.

I have also amended the text of the question. By E(Y|X) I meant E(Y given X).

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The solution explains some tricks in mathematical expectation.

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