Proving Unbiased Estimator: E(2/n*sum from 1 to n of Y(i)) = theta

[semidevil]

ok, so we know that an estimator for 1/theta, for 0<y<theta is (theta hat) = 2/n * sum from 1 to n of Y(i).


to prove that the estimator is unbiased, I need to show that the expected value of (theta hat) = theta.

so E(2/n*sum from 1 to n of Y(i)) =

2/n * sum from 1 to n of E(Y(i)).

then the book says we can cancel stuff because E(Y(i)) = theta/2.

so why is it equal to theta/2? I'm doing other problems similar to this, so do I just put E(Y(i)) = theta/2 for everything?

confused...

 

[juvenal]

I think that E(y) is equal to y/2 ONLY if the distribution of y is uniform.

Think about a flat probability distribution function in y from 0 to 1. The expected value of y is 1/2.

You also mentioned that the estimator is for 1/theta, not theta, which doesn't seem consistent since you're referring to theta^hat.

 

[thepatientmental]

To prove that an estimator is unbiased, we need to show that its expected value is equal to the true population parameter. In this case, we are trying to show that the expected value of (theta hat) is equal to theta.

First, we can rewrite (theta hat) as 2/n * sum from 1 to n of Y(i) as given in the problem. Then, using linearity of expectation, we can move the constant 2/n outside of the sum and write it as 2/n * sum from 1 to n of E(Y(i)).

Next, we need to determine the expected value of Y(i). The problem states that 0 < Y(i) < theta, which means that the distribution of Y(i) is bounded between 0 and theta. This suggests that Y(i) follows a uniform distribution with parameters 0 and theta. The expected value of a uniform distribution on the interval [a, b] is (a + b) / 2. In this case, a = 0 and b = theta, so E(Y(i)) = (0 + theta) / 2 = theta / 2.

Thus, we can rewrite our expression as 2/n * sum from 1 to n of (theta / 2). Since the sum is just adding n copies of theta / 2, we can simplify this to (2/n) * (n * theta / 2) = theta, which is the true population parameter. Therefore, we have shown that the expected value of (theta hat) is equal to theta, which proves that the estimator is unbiased.

In summary, to prove that an estimator is unbiased, we need to show that its expected value is equal to the true population parameter. In this problem, we showed that the expected value of (theta hat) is equal to theta by using the fact that Y(i) follows a uniform distribution and the properties of linearity of expectation.