STAT 230 Final, Spring 1997

American University of Beirut Math 233 Final Exam June 16, 1997 Time=1 hr 30 minutes . Let f(z) = (1/8)[1 - |(2 — @)/B|] for a- 8 <4 < a+ where a and 6 are fixed constants satisfying —00 < a < oo and 8 > 0. (a) Demonstrate that f(x) is a pdf of X.(5pts) (b) Find the cumulative distribution function (cdf) of X.(5pts) (c) Find the mean and the variance of X. (10pts) . A fair coin is tossed four times. Let X be the number of times a head is followed by tail. Find the distribution, mean and variance of X.(10pts) . Let X and Y be two random variables for which V(X) = 24, V(Y) = 54, and the correlation coefficient between X and Y, p = —0.5. What is the variance of X — Y? (10pts) . Define random variables S = X + Y and D = X — Y, where X and Y are the values obtained on the first and second rolls of a balanced die. Find the mean and variance of S and D and show that S and D are uncorrelated. Are S and D independent? Explain briefly. (10pts) . Let X and Y be independent variables, each having the same ge- ometric distribution ic. f(z) = p(l — p)? forz=0,1,2,..... Find P(X = Y).(10pts) . Suppose X and Y are independent random variables, each is uniformly distributed over the interval (0,1). Find the distribution of the two random variables / = XY and V = X/Y respectively.(20pts)