91Ó°ÊÓ

Annie and Alvie have agreed to meet between\({\rm{5:00 P}}{\rm{.M}}\). and\({\rm{6:00 P}}{\rm{.M}}\). for dinner at a local health-food restaurant. Let\({\rm{X = }}\)Annie's arrival time and\({\rm{Y = }}\)Alvie's arrival time. Suppose\({\rm{X}}\)and\({\rm{Y}}\)are independent with each uniformly distributed on the interval\({\rm{(5,6)}}\).

a. What is the joint pdf of\({\rm{X}}\)and\({\rm{Y}}\)?

b. What is the probability that they both arrive between\({\rm{5:15}}\)and\({\rm{5:45}}\)?

c. If the first one to arrive will wait only \({\rm{10\;min}}\) before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? (Hint: The event of interest is\({\rm{A = \{(x,y):|x - y|£1/6\}}}\).)

Two different professors have just submitted final exams for duplication. Let \({\rm{X}}\) denote the number of typographical errors on the first professor’s exam and \({\rm{Y}}\) denote the number of such errors on the second exam. Suppose \({\rm{X}}\) has a Poisson distribution with parameter \({{\rm{\mu }}_{\rm{1}}}\), \({\rm{Y}}\) has a Poisson distribution with parameter \({{\rm{\mu }}_{\rm{2}}}\), and \({\rm{X}}\) and \({\rm{Y}}\) are independent.

a. What is the joint pmf of \({\rm{X}}\) and\({\rm{Y}}\)?

b. What is the probability that at most one error is made on both exams combined?

c. Obtain a general expression for the probability that the total number of errors in the two exams is m (where \({\rm{m}}\) is a nonnegative integer). (Hint: \({\rm{A = }}\left\{ {\left( {{\rm{x,y}}} \right){\rm{:x + y = m}}} \right\}{\rm{ = }}\left\{ {\left( {{\rm{m,0}}} \right)\left( {{\rm{m - 1,1}}} \right){\rm{,}}.....{\rm{(1,m - 1),(0,m)}}} \right\}\)Now sum the joint pmf over \({\rm{(x,y)}} \in {\rm{A}}\)and use the binomial theorem, which says that

\({\rm{P(X + Y = m)}}\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\sum\limits_{{\rm{k = 0}}}^{\rm{m}} {\left( {\begin{array}{*{20}{c}}{\rm{m}}\\{\rm{k}}\end{array}} \right){{\rm{a}}^{\rm{k}}}{{\rm{b}}^{{\rm{m - k}}}}{\rm{ = }}\left( {{\rm{a + b}}} \right)} ^{\rm{m}}}\)

Two components of a minicomputer have the following joint pdf for their useful lifetimes \({\rm{X}}\)and \({\rm{Y}}\)

a. What is the probability that the lifetime \({\rm{X}}\) of the first component exceeds \({\rm{3}}\)?

b. What are the marginal pdf’s of \({\rm{X}}\)and \({\rm{Y}}\)? Are the two lifetimes independent? Explain.

c. What is the probability that the lifetime of at least one component exceeds\({\rm{3}}\)?

You have two lightbulbs for a particular lamp. Let\({\rm{X = }}\)the lifetime of the first bulb and\({\rm{Y = }}\)the lifetime of the second bulb (both in\({\rm{1000}}\)s of hours). Suppose that\({\rm{X}}\)and\({\rm{Y}}\)are independent and that each has an exponential distribution with parameter\({\rm{\lambda = 1}}\).

a. What is the joint pdf of\({\rm{X}}\)and\({\rm{Y}}\)?

b. What is the probability that each bulb lasts at most\({\rm{1000}}\)hours (i.e.,\({\rm{X£1}}\)and\({\rm{Y£1)}}\)?

c. What is the probability that the total lifetime of the two bulbs is at most\({\rm{2}}\)? (Hint: Draw a picture of the region before integrating.)

d. What is the probability that the total lifetime is between\({\rm{1}}\)and\({\rm{2}}\)?

Consider a system consisting of three components as pictured. The system will continue to function as long as the first component functions and either component \({\rm{2}}\) or component \({\rm{3}}\)functions. Let \({{\rm{X}}_{{\rm{1,}}}}{{\rm{X}}_{\rm{2}}}\), and \({{\rm{X}}_{\rm{3}}}\) denote the lifetimes of components \({\rm{1}}\), \({\rm{2}}\), and \({\rm{3}}\), respectively. Suppose the \({{\rm{X}}_{\rm{i}}}\) ’s are independent of one another and each \({{\rm{X}}_{\rm{i}}}\) has an exponential distribution with parameter \({\rm{\lambda }}\).

a. Let \({\rm{Y}}\) denote the system lifetime. Obtain the cumulative distribution function of \({\rm{Y}}\)and differentiate to obtain the pdf. (Hint: \({{\rm{F}}_{\left( {\rm{Y}} \right)}}{\rm{P}}\left\{ {{\rm{Y}} \le {\rm{y}}} \right\}\); express the event \(\left\{ {{\rm{Y}} \le {\rm{y}}} \right\}\)in terms of unions and/or intersections of the three events \(\left\{ {{{\rm{X}}_{\rm{i}}} \le {\rm{y}}} \right\}\), \(\left\{ {{{\rm{X}}_{\rm{2}}} \le {\rm{y}}} \right\}\), and \(\left\{ {{{\rm{X}}_3} \le {\rm{y}}} \right\}\).)

b. Compute the expected system lifetime

Answer the following questions:

a. Given that\({\rm{X = 1}}\), determine the conditional pmf of \({\rm{Y}}\)-i.e., \({{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(0}}\mid {\rm{1),}}{{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(1}}\mid {\rm{1)}}\), and\({{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(2}}\mid {\rm{1)}}\).

b. Given that two houses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island?

c. Use the result of part (b) to calculate the conditional probability\({\rm{P(Y£

1}}\mid {\rm{X = 2)}}\).

d. Given that two houses are in use at the full-service island, what is the conditional pmf of the number in use at the self-service island?

The joint pdf of pressures for right and left front tires.

a. Determine the conditional pdf of \({\rm{Y}}\) given that \({\rm{X = x}}\) and the conditional pdf of \({\rm{X}}\) given that \({\rm{Y = y}}\).

b. If the pressure in the right tire is found to be \({\rm{22}}\) psi, what is the probability that the left tire has a pressure of at least \({\rm{25}}\) psi? Compare this to.

c. If the pressure in the right tire is found to be \({\rm{22}}\) psi, what is the expected pressure in the left tire, and what is the standard deviation of pressure in this tire?

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let \({\rm{X}}\)denote the number of hoses being used on the self-service island at a particular time, and let\({\rm{Y}}\)denote the number of hoses on the full-service island in use at that time. The joint \({\rm{pmf}}\) of \({\rm{X}}\)and \({\rm{Y}}\) appears in the accompanying tabulation.

a. What is\({\rm{P(X = 1 and Y = 1)}}\)?

b. Compute P(X£1}and{Y£1)

c. Give a word description of the event , and compute the probability of this event.

d. Compute the marginal \({\rm{pmf}}\) of \({\rm{X}}\)and of \({\rm{Y}}\). Using \({{\rm{p}}_{\rm{X}}}{\rm{(x)}}\)what is P(X£1)?

e. Are \({\rm{X}}\)and\({\rm{Y}}\)independent \({\rm{rv's}}\)? Explain

Let\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}{\rm{,}}{{\rm{X}}_{\rm{4}}}{\rm{,}}{{\rm{X}}_{\rm{5}}}\), and \({{\rm{X}}_{\rm{6}}}\) denote the numbers of blue, brown, green, orange, red, and yellow M\&M candies, respectively, in a sample of size\({\rm{n}}\). Then these \({{\rm{X}}_{\rm{i}}}\) 's have a multinomial distribution. According to the M\&M Web site, the color proportions are\({{\rm{p}}_{\rm{1}}}{\rm{ = }}{\rm{.24,}}{{\rm{p}}_{\rm{2}}}{\rm{ = }}{\rm{.13}}\), \({{\rm{p}}_{\rm{3}}}{\rm{ = }}{\rm{.16,}}{{\rm{p}}_{\rm{4}}}{\rm{ = }}{\rm{.20,}}{{\rm{p}}_{\rm{5}}}{\rm{ = }}{\rm{.13}}\), and\({{\rm{p}}_{\rm{6}}}{\rm{ = }}{\rm{.14}}\).

a. If\({\rm{n = 12}}\), what is the probability that there are exactly two M\&Ms of each color?

b. For\({\rm{n = 20}}\), what is the probability that there are at most five orange candies? (Hint: Think of an orange candy as a success and any other color as a failure.)

c. In a sample of\({\rm{20M \backslash Ms}}\), what is the probability that the number of candies that are blue, green, or orange is at least \({\rm{10}}\) ?

An instructor has given a short quiz consisting of two parts. For a randomly selected student, let \({\rm{X = }}\) the number of points earned on the first part and \({\rm{Y = }}\) the number of points earned on the second part. Suppose that the joint pmf of \({\rm{X}}\) and \({\rm{Y}}\) is given in the accompanying table.

a. If the score recorded in the grade book is the total number of points earned on the two parts, what is the expected recorded score\({\rm{E(X + Y)}}\)?

b. If the maximum of the two scores is recorded, what is the expected recorded score?