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MAE 3020, Information Processing Assignment No. 3 1. The proportion of people who response to a certain mail-order solicitation is a continuous random variable X that has the density function: 2( x 2) f ( x) 5 0 0 x 1 elsewhere (a) Show that P(0 < X < 1) = 1 (b) Find the probability that more than ¼ but fewer than ½ of the people contacted will respond to this type of solicitation. 2. A shipment of 7 television sets contains 2 defective sets. A hotel makes a random purchase of 3 of the sets. If x is the number of defective sets purchased by the hotel, find the probability distribution of X. Express the results graphically as a probability histogram (frequency distribution). Furthermore, find the cumulative distribution of the random variable X representing the number of defectives as well as P(X = 1) and P(0 < X < ½). 3. The probability distribution of the discrete random variable X is: 3 1 3 f ( x) x 4 4 x 3 x , x 0,1, 2, 3 Find the mean and the variance of X. 4. Two tire quality experts examine stacks of tires and give quality ratings to each tire on a 3-point scale. Let X denote the grade given by expert A, and Y denote the grade given by B. The following table gives the joint probability for X and Y: X 1 2 3 1 0.1 0.1 0.03 Y 2 0.05 0.35 0.1 3 0.02 0.05 0.2 Find , x and y. 5. If a dealer’s profit, in units of $1000, on a new automobile can be looked upon as a random variable X having the density function: 21 x 0 x 1 f ( x) elsewhere 0 find the average profit per automobile and the variance. 6. Following the previous example, what is the dealer’s average profit per automobile if the profit on each automobile is given by g(X) = X2? Find the variance as well. 7. The random variable, X, representing the number of errors per 100 lines of software code, has the following probability distribution: x f(x) 2 0.01 3 0.25 4 0.4 5 0.3 6 0.04 Find the mean and the variance of X. 8. Suppose that a grocery store purchases 5 cartons of skim milk at the whose sale price of $1.20 per carton and retails the milk at $1.65 per carton. After the expiration date, the unsold milk is removed from the shelf and the grocer receives a credit from the distributor equal to three-fourths of the wholesale price. If the probability distribution of the random variable X, the number of cartons that are sold from this lot, is give by: x f(x) 0 1/15 1 2/15 2 2/15 3 3/15 4 4/15 5 3/15 Find the expected profit. 9. Suppose that X and Y are independent random variables having the join probability distribution: x f(x, y) y 1 3 5 2 0.1 0.2 0.1 4 0.15 0.3 0.15 Find (a) E(2X – 3Y), (b) E(XY). 10. Find the covariance of the random variable X and Y having the joint probability distribution. x y 0 x 1, 0 y 1 f ( x) elsewhere 0