12 Reconsider the situation of Examples 5.3 and 5.4 involving X5 the proportion of

Example 5.12 Reconsider the situation of Examples 5.3 and 5.4 involving X5 the proportion of

  time that a bank’s drive-up facility is busy and Y the analogous proportion for the walk-up window. The conditional pdf of Y given that X .8 is

  f (.8, y) 1.2(.8 1 y 2 )

  f Y u X (y u .8) 5

  5 5 (24 1 30y 2 ) 0,y,1

  f X (.8)

  The probability that the walk-up facility is busy at most half the time given that

  X .8 is then

  P(Y .5 u X 5 .8) 5 冮 f Y u X (y u .8) dy 5 冮 (24 1 30y 2 ) dy 5 .390

  Using the marginal pdf of Y gives P(Y .5) .350. Also E(Y) .6, whereas the expected proportion of time that the walk-up facility is busy given that X .8 (a conditional expectation) is

  1 E(Y u X 5 .8) 5 1 y f

  y (24 1 30y 冮 2

  Y X 34 ) dy 5 .574 冮 0

  If the two variables are independent, the marginal pmf or pdf in the denominator will cancel the corresponding factor in the numerator. The conditional distribution is then identical to the corresponding marginal distribution.

  EXERCISES Section 5.1 (1–21)

  1. A service station has both self-service and full-service islands.

  1 and Y 1) from the joint probability table, and verify that it equals the product P(X

  b. Compute P(X

  On each island, there is a single regular unleaded pump with

  two hoses. Let X denote the number of hoses being used on the

  P(Y

  self-service island at a particular time, and let Y denote the num-

  c. What is P(X

  Y 0) (the probability of no violations)?

  ber of hoses on the full-service island in use at that time. The

  d. Compute P(X

  Y 1).

  joint pmf of X and Y appears in the accompanying tabulation.

  3. A certain market has both an express checkout line and a

  y

  superexpress checkout line. Let X 1 denote the number of

  p(x, y)

  0 1 2 customers in line at the express checkout at a particular

  time of day, and let X 2 denote the number of customers in

  line at the superexpress checkout at the same time. Suppose

  the joint pmf of X 1 and X 2 is as given in the accompanying

  table.

  a. What is P(X

  1 and Y 1)?

  x

  b. Compute P(X

  1 and Y 1). 2

  c. Give a word description of the event {X ⬆ 0 and Y ⬆ 0}, and compute the probability of this event.

  d. Compute the marginal pmf of X and of Y. Using p X (x),

  what is P(X 1)?

  e. Are X and Y independent rv’s? Explain.

  2. When an automobile is stopped by a roving safety patrol,

  each tire is checked for tire wear, and each headlight is

  a. What is P(X 1 2 1, X 1), that is, the probability that

  checked to see whether it is properly aimed. Let X denote the

  there is exactly one customer in each line?

  number of headlights that need adjustment, and let Y denote

  b. What is P(X 1 X 2 ), that is, the probability that the numbers

  the number of defective tires.

  of customers in the two lines are identical?

  a. If X and Y are independent with p X (0) .5, p X (1) .3,

  c. Let A denote the event that there are at least two more cus-

  p X (2) .2, and p Y (0) .6, p Y (1) .1, p Y (2) p Y (3) .05,

  tomers in one line than in the other line. Express A in

  and p Y (4) .2, display the joint pmf of (X, Y) in a joint

  terms of X 1 and X 2 , and calculate the probability of this

  probability table.

  event.

  CHAPTER 5 Joint Probability Distributions and Random Samples

  d. What is the probability that the total number of customers

  b. What is the probability that there is at most one car and

  in the two lines is exactly four? At least four?

  at most one bus during a cycle?

  4. Return to the situation described in Exercise 3.

  c. What is the probability that there is exactly one car

  a. Determine the marginal pmf of X 1 , and then calculate the

  during a cycle? Exactly one bus?

  expected number of customers in line at the express

  d. Suppose the left-turn lane is to have a capacity of five

  checkout.

  cars, and that one bus is equivalent to three cars. What is

  b. Determine the marginal pmf of X 2 .

  the probability of an overflow during a cycle?

  c. By inspection of the probabilities P(X 1 4), P(X 2 0),

  e. Are X and Y independent rv’s? Explain.

  and P(X 1 4, X 2 0), are X 1 and X 2 independent random

  8. A stockroom currently has 30 components of a certain type,

  variables? Explain.

  of which 8 were provided by supplier 1, 10 by supplier 2,

  5. The number of customers waiting for gift-wrap service at a

  and 12 by supplier 3. Six of these are to be randomly

  selected for a particular assembly. Let X the number of

  department store is an rv X with possible values 0, 1, 2, 3, 4

  supplier 1’s components selected, Y the number of sup-

  and corresponding probabilities .1, .2, .3, .25, .15. A randomly

  selected customer will have 1, 2, or 3 packages for wrapping

  plier 2’s components selected, and p(x, y) denote the joint

  with probabilities .6, .3, and .1, respectively. Let Y the total

  pmf of X and Y.

  number of packages to be wrapped for the customers waiting

  a. What is p(3, 2)? [Hint: Each sample of size 6 is equally

  likely to be selected. Therefore, p(3, 2) (number of

  in line (assume that the number of packages submitted by one

  customer is independent of the number submitted by any other

  outcomes with X

  3 and Y 2)(total number of out-

  customer).

  comes). Now use the product rule for counting to obtain

  a. Determine P(X

  3, Y 3), i.e., p(3, 3).

  the numerator and denominator.]

  b. Determine p(4, 11).

  b. Using the logic of part (a), obtain p(x, y). (This can be thought of as a multivariate hypergeometric

  6. Let X denote the number of Canon digital cameras sold dur-

  distribution—sampling without replacement from a

  ing a particular week by a certain store. The pmf of X is

  finite population consisting of more than two cate-

  x

  0 1 2 3 4 gories.) 9. Each front tire on a particular type of vehicle is supposed to

  be filled to a pressure of 26 psi. Suppose the actual air pres- sure in each tire is a random variable—X for the right tire

  Sixty percent of all customers who purchase these cameras

  and Y for the left tire, with joint pdf

  also buy an extended warranty. Let Y denote the number of purchasers during this week who buy an extended

  a. What is P(X

  4, Y 2)? [Hint: This probability equals

  P(Y

  2 | X 4) P(X 4); now think of the four

  a. What is the value of K?

  purchases as four trials of a binomial experiment, with

  b. What is the probability that both tires are underfilled?

  success on a trial corresponding to buying an extended

  c. What is the probability that the difference in air pressure

  warranty.]

  between the two tires is at most 2 psi?

  b. Calculate P(X Y).

  d. Determine the (marginal) distribution of air pressure in

  c. Determine the joint pmf of X and Y and then the marginal

  the right tire alone.

  pmf of Y.

  e. Are X and Y independent rv’s?

  7. The joint probability distribution of the number X of cars

  10. Annie and Alvie have agreed to meet between 5:00 P . M . and

  and the number Y of buses per signal cycle at a proposed

  6:00 P . M . for dinner at a local health-food restaurant. Let

  X Annie’s arrival time and Y Alvie’s arrival time.

  left-turn lane is displayed in the accompanying joint

  probability table.

  Suppose X and Y are independent with each uniformly dis- tributed on the interval [5, 6].

  y

  a. What is the joint pdf of X and Y?

  p(x, y)

  0 1 2 b. What is the probability that they both arrive between 5:15 and 5:45?

  c. If the first one to arrive will wait only 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 A5 E(x, y): | x 2 y | 1 F .]

  11. Two different professors have just submitted final exams for duplication. Let X denote the number of typographical errors

  a. What is the probability that there is exactly one car and

  on the first professor’s exam and Y denote the number of

  exactly one bus during a cycle?

  such errors on the second exam. Suppose X has a Poisson

  5.1 Jointly Distributed Random Variables

  distribution with parameter m 1 , Y has a Poisson distribution

  first component functions and either component 2 or com-

  with parameter m 2 , and X and Y are independent.

  ponent 3 functions. Let X 1 ,X 2 , and X 3 denote the lifetimes

  a. What is the joint pmf of X and Y?

  of components 1, 2, and 3, respectively. Suppose the X i ’s are

  b. What is the probability that at most one error is made on

  independent of one another and each X i has an exponential

  both exams combined?

  distribution with parameter l.

  c. Obtain a general expression for the probability that the total number of errors in the two exams is m (where m is

  a nonnegative integer). [Hint: A {(x, y): x y m} {(m, 0), (m 1, 1), . . . , (1, m 1), (0, m)}. Now

  sum the joint pmf over (x, y)

  A and use the binomial

  theorem, which says that

  m

  g m a

  ba a. Let Y denote the system lifetime. Obtain the cumulative

  distribution function of Y and differentiate to obtain the pdf. [Hint: F(y) P(Y y); express the event {Y y}

  for any a, b.]

  in terms of unions andor intersections of the three events

  12. Two components of a minicomputer have the following

  {X 1 y}, {X 2 y}, and {X 3 y}.]

  joint pdf for their useful lifetimes X and Y:

  b. Compute the expected system lifetime.

  xe 2x(11y) x 0 and y 0

  16. a. For f(x 1 , x 2 , x 3 ) as given in Example 5.10, compute the f (x, y) 5 e joint marginal density function of X 1 and X 3 alone (by

  0 otherwise

  integrating over x 2 ).

  a. What is the probability that the lifetime X of the first

  b. What is the probability that rocks of types 1 and 3

  component exceeds 3?

  together make up at most 50 of the sample? [Hint: Use

  b. What are the marginal pdf’s of X and Y? Are the two life-

  the result of part (a).]

  times independent? Explain.

  c. Compute the marginal pdf of X 1 alone. [Hint: Use the

  c. What is the probability that the lifetime of at least one

  result of part (a).]

  component exceeds 3?

  17. An ecologist wishes to select a point inside a circular sam-

  13. You have two lightbulbs for a particular lamp. Let X

  the pling region according to a uniform distribution (in practice

  lifetime of the first bulb and Y

  the lifetime of the second this could be done by first selecting a direction and then a

  distance from the center in that direction). Let X the x

  bulb (both in 1000s of hours). Suppose that X and Y are

  coordinate of the point selected and Y the y coordinate of

  independent and that each has an exponential distribution

  with parameter l

  1. the point selected. If the circle is centered at (0, 0) and has

  a. What is the joint pdf of X and Y?

  radius R, then the joint pdf of X and Y is

  b. What is the probability that each bulb lasts at most

  1000 hours (i.e., X

  x 2 1y 2 R 1 and Y 1)? 2

  f (x, y) 5 u

  pR 2

  c. What is the probability that the total lifetime of the two

  bulbs is at most 2? [Hint: Draw a picture of the region

  0 otherwise

  A {(x, y): x 0, y 0, x y 2} before integrating.]

  a. What is the probability that the selected point is

  d. What is the probability that the total lifetime is between

  within R2 of the center of the circular region? [Hint:

  1 and 2?

  Draw a picture of the region of positive density D.

  14. Suppose that you have ten lightbulbs, that the lifetime of

  Because f(x, y) is constant on D, computing a proba-

  each is independent of all the other lifetimes, and that each

  bility reduces to computing an area.]

  lifetime has an exponential distribution with parameter l.

  b. What is the probability that both X and Y differ from 0 by

  a. What is the probability that all ten bulbs fail before

  at most R2?

  time t?

  c. Answer part (b) for R22 replacing R2.

  b. What is the probability that exactly k of the ten bulbs fail

  d. What is the marginal pdf of X? Of Y? Are X and Y

  before time t?

  independent?

  c. Suppose that nine of the bulbs have lifetimes that are

  18. Refer to Exercise 1 and answer the following questions:

  exponentially distributed with parameter l and that the

  1, determine the conditional pmf of Y—i.e., p Y|X (0 | 1), p Y|X (1 | 1), and p Y|X (2 | 1).

  a. Given that X

  remaining bulb has a lifetime that is exponentially dis-

  tributed with parameter u (it is made by another manu-

  b. Given that two hoses are in use at the self-service island,

  facturer). What is the probability that exactly five of the

  what is the conditional pmf of the number of hoses in use

  ten bulbs fail before time t?

  on the full-service island?

  15. Consider a system consisting of three components as pic-

  c. Use the result of part (b) to calculate the conditional

  tured. The system will continue to function as long as the

  probability P(Y

  1 | X 2).

  CHAPTER 5 Joint Probability Distributions and Random Samples

  d. Given that two hoses are in use at the full-service island,

  the color proportions are p

  1 .24, p 2 .13, p 3 .16,

  what is the conditional pmf of the number in use at the

  p

  4 .20, p 5 .13, and p 6 .14.

  self-service island?

  a. If n

  12, what is the probability that there are exactly

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

  two MMs of each color?

  given in Exercise 9.

  b. For n

  20, what is the probability that there are at most

  a. Determine the conditional pdf of Y given that X x and

  five orange candies? [Hint: Think of an orange candy as

  the conditional pdf of X given that Y y.

  a success and any other color as a failure.]

  b. If the pressure in the right tire is found to be 22 psi, what

  c. In a sample of 20 MMs, what is the probability that the

  is the probability that the left tire has a pressure of at

  number of candies that are blue, green, or orange is at

  least 25 psi? Compare this to P(Y 25).

  least 10?

  c. If the pressure in the right tire is found to be 22 psi, what

  21. Let X 1 ,X

  2 , and X 3 be the lifetimes of components 1, 2, and

  is the expected pressure in the left tire, and what is the

  3 in a three-component system.

  standard deviation of pressure in this tire?

  a. How would you define the conditional pdf of X 3 given

  20. Let X 1 , X 2 , X 3 , X 4 , X 5 , and X 6 denote the numbers of blue,

  that X 1 x 1 and X 2 x 2 ?

  brown, green, orange, red, and yellow MM candies,

  b. How would you define the conditional joint pdf of X 2 and

  respectively, in a sample of size n. Then these X i ’s have a

  X 3 given that X 1 x 1 ?

  multinomial distribution. According to the MM Web site,