0.1 Exercises for Chapter 41. In a gasoline station there are two self-service pumps and two full-service pumps.Let X denote the number of self-service pumps used at a particular time and Y thenumber of full-service pumps in use at that time. The joint pmf p(x, y) of (X, Y )appears in the next table.y0 1 20 .10 .04 .02x 1 .08 .20 .062 .06 .14 .30a) Find the probability P (X ≤ 1, Y ≤ 1).b) Compute the marginal pmf of X and Y .c) Compute the mean value and variance of X and Y .d) Compute the covariance and correlation between X and Y .2. Suppose that X and Y are discrete random variables taking values −1, 0, and 1,and that their joint pmf is given byXp(x, y) -1 0 1-1 0 1/9 2/9Y 0 1/9 1/9 1/91 2/9 1/9 0a) Find the marginal pmf of X.b) Are X and Y independent? Why?c) Find E(X).d) Find Cov(X, Y ).e) Find the probability P (X ≥ 0 and Y ≥ 0).f) Find E(max{X, Y } ).13. The following table shows the joint probability distribution of X, the amount ofdrug administered to a randomly selected laboratory rat, and Y , the number oftumors present on the rat.yp(x, y) 0 1 20.0 mg/kg .388 .009 .003 .400x 1.0 mg/kg .485 .010 .005 .5002.0 mg/kg .090 .008 .002 .100.963 .027 .010 1.000a) Are X and Y independent random variables? Explain.b) What is the probability that a randomly selected rat has: (i) one tumor, and (ii)at least one tumor?c) For a randomly selected rat in the 1.0 mg/kg drug dosage group, what is theprobability that it has: (i) no tumor, (ii) at least one tumor?d) What is the expected number of tumors for a randomly selected rat in the 1.0mg/kg drug dosage group?e) Does the 1.0 mg/kg drug dosage increase or decrease the expected number oftumors over the 0.0 mg/kg drug dosage? Justify your answer.f) You are given: E(X) = .7, E(X2) = .9, E(Y ) = .047, E(Y2) = .067. Find: (i)Cov(X, Y ), and (ii) ρX,Y.4. Let X be defined by the probability density functionf(x) =−2x −1 < x ≤ 02x 0 < x ≤ 10 otherwisea) Find E(X3).b) Define Y = X2and find cov(X, Y ).c) Are X and Y independent? Why or why not?5. Let X and Y be defined by the joint probability density function given below:2f(x, y) =(2e−x−y0 ≤ x ≤ y < ∞0 otherwisea) Find P (X + Y ≤ 3).b) Find the marginal pdf’s of Y and X.b) Are X and Y independent? Justify your answer.6. Let X take the value 0 if a child under 5 uses no seat belt, 1 if it uses adult seatbelt, and 2 if it uses child seat. And let Y take the value 0 if a child survived amotor vehicle accident, and 1 if it did not. An extensive study undertaken by theNational Highway Traffic Safety Administration resulted in the following conditionaldistributions of Y given X = x:y | 0 1------------------------P(Y=y|x=0) | .69 .31------------------------P(Y=y|x=1) | .85 .15------------------------P(Y=y|x=2) | .84 .16while the marginal distribution of X isx | 0 1 2-------------------------P(X=x) | .54 .17 .29(a) Use the table of conditional distributions of Y given X = x to conclude whetheror not X and Y independent. Justify your answer.(b) Tabulate the joint distribution of X and Y .(c) Use the joint distribution of X and Y to conclude whether or not X and Y areindependent. Justify your answer.(d) Find the marginal distribution of Y .(e) Find the regression function, µY |X(x), of Y on X.(f) Find the covariance and the correlation of X and Y .37. A popular local restaurant offers dinner entrees in two price ranges: $7.50 meals and$10.00 meals. From past experience, the restaurant owner knows that 65% of thecustomers order from the $7.50 meals, while 35% order from the $10.00 meals. Alsofrom past experience, the waiters and waitresses know that the customers always tipeither $1.00, $1.50, or $2.00 per meal. The table below gives the joint probabilitydistribution of the price of the meal ordered and the tip left.Tip Left$1.00 $1.50 $2.00Meal Price $7.50 .455 .195 0 .65$10.00 .035 .210 .105 .351(a) Find the correlation between the price of the meal and the amount of the tipleft. You may use the following information to help you answer the question:E(price) = 8.38, V ar(price) = 1.42, E(tip) = 1.31, and V ar(tip) = 0.11.(b) Is the amount of tip left independent of the price of the meal? Justify youranswer.(c) Find the regression function E(Tip|Meal Price).8. Consider selecting two products from a batch of 10 products. Suppose that thebatch contains 3 defective and 7 non-defective products. Let X take the value 1or 0 as the first selection from the 10 products is defective or not. Let Y take thevalue 1 or 0 as the second selection (from the nine remaining products) is defectiveor not.(a) Find the marginal distribution of X.(b) Find the conditional distributions of Y given each of the possible values of X.(c) Use your two results above to find the joint distribution of X and Y .(d) Find the regression function E(Y |X = x), x = 0, 1.(e) Find the conditional variance of Y given X = 1.(f) Find the marginal distribution of Y . Is it the same as that of X?(g) Find the covariance and the correlation of X and Y .49. Suppose your waiting time for the bus in the morning has mean 3 minutes andvariance 1.12 minutes2, while the waiting time in the evening has mean 6 minutesand variance 4 minutes2. In a typical week, you take the bus 5 times in the morningand 3 times in the evening.(a) Calculate the exp ected value of the total waiting time in a typical week.(b) Calculate the variance of the total waiting time in a typical week. State yourassumptions.(Hint: Let Xidenote the waiting time in the ith morning of the week, i = 1, . . . , 5,and let Yjdenote the waiting time in the jth evening of the week. Express the totalwaiting as a linear combination of the random variables X1, . . . , X5, Y1, Y2, Y3.)10. Two towers are constructed, each by stacking 30 segments of concrete vertically.The height (in inches) of a randomly selected segment is uniformly distributed inthe interval (35.5,36.5).(a) Find the mean value and variance of the height of a randomly selected segment.(Hint: What is the mean and variance of a uniform in (35.5,36.5) randomvariable?)(b) Let X1, . . . , X30denote the heights of the segments used in tower 1, andY1, . . . , Y30denote the heights of the segments used in tower 2. Express thedifference of the heights of the two towers in terms of these Xs and Y s. (Hint:Height of tower 1 is the sum of the Xs.)(c) Find the mean and variance of the difference of the heights of the two towers.11. Consider the following situation from architectural design. The