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24.5. Exercises

1. Let \(X\) and \(Y\) be jointly distributed random variables and let \(\hat{Y}\) be the linear regression estimate of \(Y\) based on \(X\). Show that the mean squared error of this estimate is \((1 - r^2)Var(Y)\) where \(r\) is the correlation between \(X\) and \(Y\). This leads to the Data 8 formula for the SD of the residuals in simple linear regression.

[Use Alternative Form I of the regression equation, and preserve deviations as we did here.]

2. Let \(X\) and \(Y\) be standard bivariate normal with correlation \(\rho\).

(a) Suppose I ask you for the least squares estimate of \(Y\) based on \(X\), but I don’t tell you \(X\). What is your estimate, and what is its mean squared error?

(b) Suppose I now show you \(X\). Now what is your least squares estimate of \(Y\), and what is its mean squared error?

(c) What is your least squares estimate of \(Y\) based only on linear functions of \(X\), and what is its mean squared error?

3. Let \((X, Y)\) be the weight and height of a person picked at random from a population, and suppose the distribution of \((X, Y)\) is bivariate normal with correlation 0.6. Suppose also that

  • \(X\) has mean 150 pounds and SD 25 pounds

  • \(Y\) has mean 68 inches and SD 3 inches

Sketch the conditional density of \(Y\) given \(X = 170\) pounds. Mark the numerical values of the conditional mean and SD appropriately on your sketch.

4. Let \(X\) and \(Y\) have a bivariate normal distribution (not necessarily standard) with correlation \(\rho \in (0, 1)\). Suppose you are given that \(X\) is on the 30th percentile.

(a) Pick the right option for the least squares estimate of \(Y\), and explain.

(i) Below the 30th percentile

(ii) On the 30th percentile

(iii) Above the 30th percentile

(b) Write a single math expression for the percentile rank corresponding to the least squares estimate of \(Y\). Your answer can involve \(\rho\) and the standard normal cdf \(\Phi\).

5. Let \(X\) and \(Y\) be standard bivariate normal with correlation \(\rho \in (0, 1)\).

(a) Without calculation, pick the right option and explain. \(P(X > 0, Y < 0)\) is

(i) less than \(0.25\)

(ii) equal to \(0.25\)

(iii) greater than \(0.25\)

(b) Now find \(P(X > 0, Y < 0)\) in terms of \(\rho\).

[No integration is needed. Write \(Y\) in terms of \(X\) and standard normal \(Z\) independent of \(X\), sketch the region, and use what you know about the joint density of \((X, Z)\).]

6. Let \(X\) and \(Y\) be standard bivariate normal with correlation \(\rho\). Find \(E(\max(X, Y))\). The easiest way is to use the fact that for any two numbers \(a\) and \(b\), \(\max(a, b) = (a + b + \vert a - b \vert)/2\). Check the fact first, and then use it.

7. Suppose that \(X\) is normal \((\mu_X, \sigma_X^2\)), \(Y\) is normal \((\mu_Y, \sigma_Y^2)\), and the two random variables are independent. Let \(S = X+Y\).

(a) Find the conditional distribution of \(X\) given \(S=s\).

(b) Find the least squares predictor of \(X\) based on \(S\) and provide its mean squared error.

(c) Find the least squares linear predictor of \(X\) based on \(S\) and provide its mean squared error.