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).

Solution Preview

Please see the attached file.

Solution
Given that
Let and
Now
...

Solution Summary

The solution explains some tricks in mathematical expectation.

... skills, the expectation of students using problem solving skills, and an expectation that encourages interaction and mathematical communication between ...

Math for Decision Making. ... The solution gives complete details of computing probability and expectation for binomial and Poisson random variables. ...

... These expectations reflect what the school needs to successfully help all students ... call them proficient and advanced, are small in reading and math, the school ...

... from some unknown distribution, then the sample mean is expressed as: <Xn>=(x1+x2+ x3+....+xn)/n The mathematical expectation of sample mean is equal to: E[<Xn ...

... Having mathematical standards in grades K-12 will give students clear and realistic expectations of what they are expected to learn in mathematics. ...

... a standard integral and one can find its value from mathematical handbooks. It is useful to remember it. (b) Since n = 17 is odd, required expectation value is ...

... to have an expected value of 0? Provide an example of expectation and the ... the assessment of evidence that are not always closely related to mathematical ability ...