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

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…

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?

Suppose that Y₁ is a negative binomial random variable with p= .75 and r = 6, Y₂ is a random variable with mean -5 and variance 11, and Y3 is a gamma random variable with 1.6 and a = 5.9. Further suppose that all of these random variables are independent. (a) What is E[2Y₁ − 6Y2 + 5Y₁Y3]? (b) What is the variance of 3Y₁ +5Y2 - 8Y3? (c) What is the mgf of 7Y₁ +4Y3? 0 =

Suppose X and Y are independent. X has a mean of 1 and variance of 1, Y has a mean of 0, and variance of 2. Let S=X+Y, calculate E(S) and Var(S). Let Z=2Y^2+1/2 X+1 calculate E(Z). Hint: for any random variable X, we have Var(X)=E(X-E(X))^2=E(X^2 )-(E(X))^2, you may want to find EY^2  with this. Calculate cov(S,X). Hint: similarly, we have cov(Z,X)=E(ZX)-E(Z)E(X), Calculate cov(Z,X). Are Z and X independent? Are Z and Y independent? Why? What about mean independence?

2. Let x1,x2,...,xn be a random sample from N(0, 2),we wish to estimate the variance σ² as: ²="Is this un unbiased estimator? Find the variance of ²? "

The random variable X is distributed normally with mean ux and variance 6, and the random variable Y is normally distributed with mean 8 and variance of.. 2X – 3Y is distributed normally with mean 12 and variance 42. Calculate the values of µx and oy respectively. Assume independence! a. Hx= - 6 and oy=2 O b.Hx = 18 and oy=/2 OC Hx = 6 and oy=/2 O d. Hx= 12 and oy=V42