STAT 230 Final, Fall 2007

AMERICAN UNIVERSITY of BEIRUT STAT 230, Final Examination Time = 1 hour and 30 minutes Jan 25, 2008 You are allowed to use a calculator, one formula sheet, tables of discrete and continuous distributions. (GOOD LUCK!) 1. Consider the experiment of rolling a pair of fair 4-sided dice. Let X1 be the outcome on die 1 , and X2 be the outcome on die 2 respec- tively. Define Y1 = max{X1, X2} and Y2 = |X1 − X2|. Determine the Covariance of Y1 and Y2. [15 pts] 2. Let the joint pdf of the two random variables X and Y be f(x, y) = kx if 0 < x < y < 1, 0 elsewhere, and k is a positive constant. (a) Find the value of k such that f(x, y) is a joint pdf. [5 pts] (b) Determine the marginal pdf of X, f1(x), and the marginal pdf of Y , f2(y), respectively. [10 pts] (c) Are the random variables X and Y independent? [5 pts] 3. Let X1 and X2 be a random sample of size 2 from the standard normal distribution. Let Y1 = (X1 + X2)/ √ 2 and Y2 = (X1 − X2)/ √ 2. Show that Y1 and Y2 are independent standard normal random variables. Hint: Try to find the joint mgf of (Y1, Y2) = E(e{t1Y1+t2Y2}). [15 pts] 4. Let X1 and X2 be a random sample from the exponential distribution with common pdf f(x) = e−x if x > 0. Let Y1 = X1/X2 and Y2 = X1 + X2. (a) Find the joint distribution of (Y1, Y2). [10 pts] (b) Find the mean and variance of Y1. [10 pts] 1 5. The ”Fill” problem is important in many industries, like those making cereal, tooth paste and so on. If such an industry claims that it is selling 100 grams of its product in a container or else the FDA will crack down. Although the FDA will allow a very few percentage to have a fill of less than 100 grams. (a) If the contents X of a container has a normal distribution with mean µ = 100.01 and an unknown variance of σ2. Find the value of σ2 if P(X < 100) = 0.025 [10 pts] (b) In 50 independent such measurements, approximate the proba- bility that at least one container is under 100 grams of fill. [5 pts] 6. In the casino game of Craps, the probability of winning a bet is 0.493. Let Y denote the number of winnings in a series of 1000 bets that are placed. Approximate P(Y > 500). [15 pts] 2