1 let x1 and x2 be independent random variables with mean...

Let X1 and X2 be independent random variables with mean µ and variance σ^2. Suppose that we have two estimators of µ: Y1 = (X1 + X2)/3 Y2 = (2X1 - X2)/3 Calculate the MSE for each estimator. Select a better estimator when µ = 3 and variance σ^2 = 0.5.

Transcript

00:01 Hi, i'm david and i'm here to help you and change your question.

00:03 Now let me bring up your question here.

00:06 In the question here, we are going to discuss about the mean square error of the estimator.

00:12 We know that the mean square error is equal to the variance of the estimator, then we plus one the bias of the estimator is totally square.

00:26 And we have the bias of the estimator, it will equal to the e of the theta heart with minus with the theta.

00:36 Now we have two questions, so let me split into the two screens, to the two equal parts.

00:43 Now the first estimate the y1, it will equal to the x1 plus x2 over 3.

00:53 And we are given the mean and the variance here.

00:57 So first of the form, for this one, i have to find e of the y1.

01:02 Equal to the e of the x1 plus the x2 over 3 and then get equal to the out of 3 e on the x1 plus e on the x2 and because they are the same so therefore we will have mean we equal to the 3 plus 3 and equal to the divided by 3 and then then get equal to the 3 and therefore this one here sorry it will be 6 divided be 3 equal to the 2.

01:35 And from here we can find the bias of the y1 equal to the 2.

01:42 The 2 will minus the mean will be the 3 and then we get equal to the minus 1.

01:47 And also we have to find the variance of the y1, just equal to the variance of the x1 plus the x2, divided by 3.

01:59 And then we get equal to we're born to bring the constant outside the variance.

02:03 We need to square.

02:04 Up so will be the three square variance of the x1 plus the variance of the x2 because they're independent and then we get equal to variance equal to the zbon 5 square plus 0 .05 square over the 9.

02:21 Then we have 0 .25 times 2 equal to the 715...

Question

Let X1 and X2 be independent random variables with mean µ and variance σ^2. Suppose that we have two estimators of µ: Y1 = (X1 + X2)/3 Y2 = (2X1 - X2)/3 Calculate the MSE for each estimator. Select a better estimator when µ = 3 and variance σ^2 = 0.5.

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