2.6.4 Problems 2.6.1 Suppose that the random variables X, Y, and Z are independent with E(X) = 2, Var(X) = 4, E(Y) = -3, Var(Y)=2, E(Z) = 8, and Var(Z) = 7. Calculate the expectation and variance of the following random variables. (a) 3X+7 (c) 2X+6Y (e) 5X 9Z+8 - (g) X+2Y+3Z (b) 5X-9 (d) 4X - 3Y (f)-3Y-Z-5 (h) 6X+2Y-Z +16

1. Random variable X has p.d.f be 6z I> 0 fx(z) = || otherwise and let Y = eX. (b) (i.e. Calculate Var(Y)) Calculate the variance of Y.

6. Show that 1 s2 E1(Xi – x)² is unbiased estimator of the population variance o? i=1 п-1

3. A research assistant is asked to draw two independent normally distributed samples. The first sample, X has a 10 observations with a mean of 1 and sample variance of 2. The second sample, Y has 20 observations with a mean of 0.5 and a sample variance of 3. Test the following hypotheses: (a) X = Ý (b) X > Y (c) Var(X) Var(Y) %3D (d) Var(X) = 1.5 (e) Var(Y) = 2.5 %3D

4. Let X be a discrete random variable with E (X) = 3 and Var(X) = 0.5, What are the mean %3D and variance of h(X) = -4x + 1? A. mean is 11, variance is 8. B. mean is –11, variance is 8. C. mean is –11, variance is 2. D. mean is 11, variance is 2. 5. Suppose that 12% of microprocessors are defective. Six microprocessors are randomly chosen. Assume that they function independently. What is the probability that they all work? A. 0.24 or 24% B. 0.44 or 44% C. 0.72 or 72% D. 0.46 or 46% 6. Which of the following standard normal probabilities are not correct? A. P(Z s-1.25) = .8944 B. P(Z >1.25) =.1056 C. P(Z <1.25) = .8944 D. P(-1.25 sZ 31.25) = .7888

2. [7+3+5+10] Consider the following model where et is i.i.d and |p| < 1. Yt = Bo+B1xt + ut Ut = put-1 + et, t = 1, 2, … … · ‚T, (a) Show that variance of the usual OLS estimator (0² / Σ(x+ - x)²) underestimates the true variance of the OLS estimator. (b) The value of the Durbin-Watson statistic turned out to be 1.4. Compute an estimate of p, the serial correlation parameter, based on the value of the Durbin-Watson statistic. (c) Suppose you also wish to test for serial correlation using Durbin's alternative test (Durbin's h- test) at the 5 percent significance. After estimating the original regression on the entire 80 sam- ples, you saved the residuals and estimated the following auxiliary regression. Et = 0.07 + 0.08æt+ 0.3 êt_1 (0.04) (0.02) (0.1) (Standard errors are in parentheses.) Give the value of the test statistic, and your conclusion (accept or reject the null hypothesis of no serial correlation). (d) Regardless of your answers to parts (a)-(c), suppose you believe that…

7. Calculate the variance of g (X) = 2X + 3, where X is a random variable with probability distribution X 0 1 2 3 f(x)=1/ 4 1180 31100 1 2 8 First, we find the mean of the random variable 2X + 3.

Suppose X and Y are two independent variables with variance 1. Let Z = X+bY where b > 0. If Cor(Z, Y ) = 1/2, what is the value of b?

3. An individual who has automobile insurance from a certain company is ran- domly selected. Let Y be the number of moving violations for which the individual was cited during the last 3 years. The pmf of Y is 1 p(y) | .60 .25 .10 .05 What is the variance of Y? Α. 1.1 В. .60 C. .79 D. 1 E. None of the above 4. Based on the pmf in problem 3, what is the probability that an individual was cited between 1 and 3 times (inclusive)? A. .25 В. .05 С. .30 D. .40 E. None of the above

If a random sample of size n is selected withoutreplacement from the finite population that consists ofthe integers 1, 2, ... , N, show that(a) the mean of X is N + 12 ;(b) the variance of X is (N + 1)(N − n)12n ;(c) the mean and the variance of Y = n · X areE(Y) = n(N + 1)2and var(Y) = n(N + 1)(N − n)12