Show that var equivalent the variance the prediction error

Solved Step by Step With Explanation-Stats of Prediction Properties

Questions

(b) Calculate the variance of Ý.

(c) Show that (b) is equivalent to the variance of the prediction error, Y-Y

To calculate the statistical properties of predictions, we'll work with the following variables:

Y: Predicted value

Let's address each part of your question:

(a) Calculate the expected value of a predicted value Y, where Y = β₀ + X, and X is the new value of X.

Var[Ŷ] = Var[β₀ + X]

(c) Show that (b) is equivalent to the variance of the prediction error, ε = Y* - Ŷ.

This equation shows that the variance of the prediction error ε is the sum of the variances of Y* and Ŷ. Thus, Var[Ŷ] is equivalent to the variance of the prediction error ε if Y* and Ŷ are independent.

(d) Show that the expected value of the mean response prediction E[Ŷ] is equivalent to (a), but Var[Ŷ] is smaller than (b).

We have shown the relationships between the expected value of predicted values (Y and Ŷ), the variance of Ŷ, and the variance of the prediction error ε, as well as the comparisons between them.