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

2. A random sample X₁, X2, X3 of size 3 is drawn from a population with mean and variance o2. Let 1 3 Đi (Xi+X+X3) - (X₁ + X2 + X3), (3X₁ — X₂ +2X3). - 4 (a) Show that ₁ and ₂ are both unbiased estimators of the population 2 = mean . (b) Compute the variance of ₁ and 2. (c) Which estimator would you use? Explain your answer.

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. A random variable X has only two values a and b with P(X = a) = p , P(X = b) = q (p+ q = 1). Find its mean value and variation.

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.

Two random variables X and Y have means X = 1 and Y=2, variances o=4 and o X and Pxy = 0.4. New random variables W and V are defined by V =−X+2Y and W = X +3Y. Find (a) the means, (b) the variances, = 1,

Check the linearity, casualty, stability and the time variance of the following: 1. y(n) = 4 x(n-2) + 2x(n-l) + x(n/2) 2. y(n) - l/n x(n+4) 3. y(n) W 0.25 x(n-1) + 0.5 x(n+1) - 0.4 y(n-1)

If x and y are independent random variable with E(xy)= 20 then the correlation coefficient is