0 1 Exercises for Chapter 4 1 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 the number of full service pumps in use at that time The joint pmf p x y of X Y appears in the next table y 0 1 2 0 10 04 02 08 20 06 2 06 14 30 x 1 a 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 by X p x y 1 0 1 1 0 1 9 2 9 0 1 9 1 9 1 9 1 2 9 1 9 0 Y a 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 1 3 The following table shows the joint probability distribution of X the amount of drug administered to a randomly selected laboratory rat and Y the number of tumors present on the rat y p x y x 0 1 2 0 0 mg kg 388 009 003 400 1 0 mg kg 485 010 005 500 2 0 mg kg 090 008 002 100 963 027 010 1 000 a 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 the probability 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 0 mg kg drug dosage group e Does the 1 0 mg kg drug dosage increase or decrease the expected number of tumors over the 0 0 mg kg drug dosage Justify your answer f You are given E X 7 E X 2 9 E Y 047 E Y 2 067 Find i Cov X Y and ii X Y 4 Let X be defined by the probability density function 2x 1 x 0 f x 2x 0 x 1 0 otherwise a Find E X 3 b Define Y X 2 and 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 2 f x y 2e x y 0 x y 0 otherwise a 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 seat belt and 2 if it uses child seat And let Y take the value 0 if a child survived a motor vehicle accident and 1 if it did not An extensive study undertaken by the National Highway Traffic Safety Administration resulted in the following conditional distributions 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 16 while the marginal distribution of X is x 0 1 2 P X x 54 17 29 a Use the table of conditional distributions of Y given X x to conclude whether or 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 are independent Justify your answer d Find the marginal distribution of Y e Find the regression function Y X x of Y on X f Find the covariance and the correlation of X and Y 3 7 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 the customers order from the 7 50 meals while 35 order from the 10 00 meals Also from past experience the waiters and waitresses know that the customers always tip either 1 00 1 50 or 2 00 per meal The table below gives the joint probability distribution of the price of the meal ordered and the tip left Tip Left 1 00 1 50 2 00 Meal Price 7 50 455 195 0 65 10 00 035 210 105 35 1 a Find the correlation between the price of the meal and the amount of the tip left 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 your answer c Find the regression function E Tip Meal Price 8 Consider selecting two products from a batch of 10 products Suppose that the batch contains 3 defective and 7 non defective products Let X take the value 1 or 0 as the first selection from the 10 products is defective or not Let Y take the value 1 or 0 as the second selection from the nine remaining products is defective or 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 4 9 Suppose your waiting time for the bus in the morning has mean 3 minutes and variance 1 12 minutes2 while the waiting time in the evening has mean 6 minutes and variance 4 minutes2 In a typical week you take the bus 5 times in the morning and 3 times in the evening a Calculate the expected value of the total waiting time in a typical week b Calculate the variance of the total waiting time in a typical week State your assumptions Hint Let Xi denote the waiting time in the ith morning of the week i 1 5 and let Yj denote the waiting time in the jth evening of the week Express the total waiting 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 …