Final answer:
The expression x-y is an unbiased estimator of 0, assuming that y is a perfect prediction of x and there's no error in prediction, as E(x-y) is calculated as E(x)-E(y) which ultimately equates to 0.
Step-by-step explanation:
To show that x-y is an unbiased estimator of a certain parameter, we need to use the rules of expected value, also known as the mean or expected value. The expected value of a random variable X is denoted as E(X), which represents the long-term average if an experiment is repeated many times. If x is an independent variable and y is a dependent variable such as in a regression setting, we assume there's an underlying true relationship, possibly y = βx + ε, where ε is the error term with E(ε) = 0. If we consider x as the third exam score and y as the final exam score, and we assume that the true relationship is such that the final exam score is perfectly predicted by the third exam score without error, then y would equal the expected value of the final exam score given x.
Now, the expected value of x-y, using linearity of expectation, is E(x-y) = E(x) - E(y). If E(y) is equal to x (since we assume y is perfectly predicted by x in our theoretical model), then E(x-y) = E(x) - x = 0, meaning we expect no difference on average if y is a perfect prediction of x. Therefore, x-y would be an unbiased estimator of 0, indicating that on average, the estimator hits the true difference, which is zero in this case.