CHAPTER 7 RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS MULTIPLE CHOICE QUESTIONS

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1 CHAPTER 7 RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer.. The weighted average of the possible values that a random variable X can assume, where the weights are the probabilities of occurrence of those values, is referred to as the: a. variance b. standard deviation c. epected value d. covariance c. The number of accidents that occur annually on a busy stretch of highway is an eample of: a. a discrete random variable b. a continuous random variable c. a discrete probability distribution d. a continuous probability distribution a. A function or rule that assigns a numerical value to each simple event of an eperiment is called: a. a sample space b. a probability tree c. a probability distribution d. a random variable d 4. If X and Y are any random variables, which of the following identities is not always true? a. E (X+Y) = E(X) + E(Y) b. V(X+Y) = V(X) + V(Y) c. E(4X+5Y) = 4E(X) + 5 E(Y) d. V(4X+5Y) = 6V(X) + 5V(Y) + 40COV(X,Y) b 5. If X and Y are random variables, the sum of all the conditional probabilities of X given a specific value of Y will always be: a

2 95 Chapter Seven b..0 c. the average of the possible values of X d. a value larger than zero but smaller than.0 b 6. A statistical measure of the strength of the linear relationship between two random variables X and Y is referred to as the: a. epected value b. variance c. covariance d. standard deviation c 7. A table, formula, or graph that shows all possible values a random variable can assume, together with their associated probabilities is called a: a. discrete probability distribution b. continuous probability distribution c. bivariate probability distribution d. probability tree a 8. If X and Y are random variables with E(X) = 5 and E(Y) = 8, then E(X+Y) is: a. 4 b. c. 8 d. 40 a 9. If X and Y are random variables with V(X) = 7.5, V(Y) = 6 and COV(X,Y) = 4, then V(X+Y) is: a. b. 7 c. 88 d. d 0. If X and Y are independent random variables, which of the following identities is always true? a. E(X+Y) = E(X) + E(Y) + 5 b. V(X+Y) =V(X) + V(Y) c. V(X+Y) = 4V(X) + 9V(Y) d. E(X+Y) = 5E(X+Y) c. If X and Y are any random variables with E(X) = 50, E(Y) = 6, E(XY) =, V(X) = 9 and V(Y) = 0, then the relationship between X and Y is a : a. strong positive relationship b. strong negative relationship c. weak positive relationship d. weak negative relationship

3 Random Variables and Discrete Probability Distributions 96 b. Which of the following is not a characteristic of a binomial eperiment? a. There is a sequence of identical trials b. Each trial results in two or more outcomes. c. The trials are independent of each other. d. Probability of success p is the same from one trial to another. b. The epected value, E(X), of a binomial probability distribution with n trials and probability p of success is: a. n + p b. np(-p) c. np d. n + p - c 4. The Poisson random variable is a: a. discrete random variable with infinitely many possible values b. discrete random variable with finite number of possible values c. continuous random variable with infinitely many possible values d. continuous random variable with finite number of possible values a 5. Given a Poisson random variable X, where the average number of successes occurring in a specified interval is.8, then P(X = 0) is a..8 b..46 c d c 6. Which probability distribution is appropriate when the events of interest occur randomly, independently of one another, and rarely? a. Binomial distribution b. Poisson distribution c. Any discrete probability distribution d. Any continuous probability distribution b 7. The epected number of heads in 00 tosses of an unbiased coin is a. 0 b. 40 c. 50 d. 60 c 8. Which of the following cannot generate a Poisson distribution?

4 97 Chapter Seven a. The number of children watching a movie b. The number of telephone calls received by a switchboard in a specified time period c. The number of customers arriving at a gas station in Christmas day d. The number of bacteria found in a cubic yard of soil a 9. The Poisson probability distribution is used with a. a discrete random variable b. a continuous random variable c. either a discrete or a continuous random variable, depending on the mean d. either a discrete or a continuous random variable, depending on the sample size a 0. The variance of a binomial distribution for which n = 00 and p = 0.0 is: a. 00 b. 80 c. 0 d. 6 d. Which of the following is (are) required condition(s) for the distribution of a discrete random variable X that can assume values i? a. b. 0 p ( i ) for all i p( ) = i all i c. Both a and b are required conditions d. Only b is a required condition. c Which of the following is not a required condition for the distribution of a discrete random variable X that can assume values i? a. b. c. 0 p ( i ) for all i p( ) = i all i p ( i ) > for all i d. All of the above are not required conditions c. The binomial probability distribution is used with a. a discrete random variable b. a continuous random variable c. either a discrete or a continuous random variable, depending on the variance

5 Random Variables and Discrete Probability Distributions 98 d. either a discrete or a continuous random variable, depending on the sample size a 4. Twenty five percent of the students in an English class 0f 00 are international students. The standard deviation of this binomial distribution is a. 5 b. 5 c d. 4. d 5. In the notation below, X is the random variable, c is a constant, and V refers to the variance. Which of the following laws of variance is not correct? a. V(c) = 0 b. V(X + c) = V(X) c. V(X + c) = V(X) + c d. V(cX) = c V(X) c TRUE/FALSE QUESTIONS 6. A random variable is a function or rule that assigns a number to each outcome of an eperiment. T 7. The time required to drive from Detroit to Chicago is a discrete random variable. F 8. The binomial distribution deals with consecutive trials, each of which has two possible outcomes. T 9. The number of home insurance policy holders is an eample of a discrete random variable T 0. The mean of a discrete probability distribution is given by the equation µ = p ( ). T. Poisson distribution is appropriate to determine the probability of a given number of defective items in a shipment. F. The length of time for which an apartment in a large comple remains vacant is a discrete random variable. F

6 99 Chapter Seven. The number of homeless people in New York City is an eample of a discrete random variable. T 4. Given that X is a discrete random variable, then the laws of epected value and variance can be applied to show that E(X + 5) = E(X), and V(X+5) = V(X) + 5. F 5. In a southern state, stabbings of inmates by other inmates at a state prison have averaged.5 per month over the past year. Let X be the number of inmate stabbings per month, and assume that a Poisson distribution is appropriate for this situation. Then, P(X = ) = 0.8. F 6. The Poisson distribution is applied to events for which the probability of occurrence over a given span of time, space, or distance is very small. T 7. For a given probability of success p, the binomial distribution tends to take on more of a bell shape as the number of trials n increases. T 8. A table, formula, or graph that shows all possible values a random variable can assume, together with their associated probabilities, is referred to as discrete probability distribution. T 9. Any discrete distribution is appropriate when the events of interest occur randomly, independently of one another, and rarely. F 40. The Poisson random variable is a discrete random variable with infinitely many possible values. T 4. The mean of a Poisson distribution, where µ is the average number of successes occurring in a specified interval, is µ. T 4. The number of accidents that occur at a busy intersection is an eample of a Poisson random variable. T 4. The variance of a binomial distribution for which n = 50 and p = 0.0 is 8.0. T 44. The epected number of heads in 50 tosses of an unbiased coin is 5. T

7 Random Variables and Discrete Probability Distributions The binomial random variable is the number of successes that occur in a period of time or an interval of space. F 46. If X is a binomial random variable with n = 5, and p = 0.5, then P(X = 5) =.0. F 47. The number of students that use a computer lab during one day is an eample of either a continuous or a discrete random variable, depending on the number of the students. F 48. The Poisson probability distribution is a continuous probability distribution. F 49. The binomial probability distribution is a discrete probability distribution. T 50. The standard deviation of a binomial random variable X is given by the formula σ = np( p ), where n is the number of trials, and p is the probability of success. F 5. The temperature of the room in which you are taking this test is a continuous quantitative variable. T TEST QUESTIONS 5. For each of the following random variables, indicate whether the variable is discrete or continuous, and specify the possible values that it can assume. a. X = the number of traffic accidents in Phoeni on a given day. b. X = the amount of weight lost in a month by a randomly selected dieter. c. X = the average number of children per family in a random sample of 00 families. d. X = the number of households out of 0 surveyed that own a microwave oven. e. X = the time in minutes required to obtain service in a restaurant. a. discrete; = 0,,,,... b. continuous; < < c. continuous; 0 d. discrete; = 0,,,..., 0 e. continuous; > 0

8 0 Chapter Seven QUESTIONS 5 THROUGH 56 ARE BASED ON THE FOLLOWING INFORMATION: The probability distribution of a discrete random variable X is shown below. p() Find the following probabilities: a. P(X > ) b. P(X ) c. P( X ) d. P(0 < X < ) e. P( X<) a. 0.5 b c d e Find the epected value, the variance, and the standard deviation of X. E(X) =.5, V(X) = , and σ = a. Find E( X ) b. Find E( X + 5) c. Find E ( X ) a..55 b. 0. c a. Find E(X-) b. Find V(X-) c. Find E( X + X - ) a..75 b c QUESTIONS 57 AND 58 ARE BASED ON THE FOLLOWING INFORMATION:

9 Random Variables and Discrete Probability Distributions 0 Consider a random variable X with the following probability distribution: p() = 0.05, 57. =,, 4, 5, or 6 Epress the probability distribution in tabular form. p() Find the following probabilities: a. P(X 4) b. P(X > 4) c. P( X 5) d. P( < X < 4) e. P(X = 4.5) ANSWERS a b c d. 0.5 e Determine which of the following are not valid probability distributions, and eplain why not. a. p() p() p() b. c. 0.0 Table (a) is not a valid probability distribution because the probabilities don t sum to one, and Table (b) is not valid because it contains a negative probability. Table (c) is a valid probability distribution.

10 0 Chapter Seven QUESTIONS 60 THROUGH 6 ARE BASED ON THE FOLLOWING INFORMATION: The probability distribution of a random variable X is shown below. p() Find the following probabilities: a. P(X 0) b. P(X > ) c. P(0 X 4) d. P(X = 5) a b c d a. Find E(X). b. Find V(X). a..40 b Find the epected value of Y = X - X a. b. Find E(X-4). Find V(X-4). a. 6.0 b QUESTIONS 64 THROUGH 66 ARE BASED ON THE FOLLOWING INFORMATION: Let X represent the number of children in an Egyptian household. The probability distribution of X is as follows: p() What is the probability that a randomly selected Egyptian household will have

11 Random Variables and Discrete Probability Distributions 04 a. more than children? b. between and 5 children? c. fewer than 4 children? a. 0.5 b. 0.4 c Find the epected number of children in a randomly selected Egyptian household Find the standard deviation of the number of children in an Egyptian household..844 QUESTIONS 67 THROUGH 70 ARE BASED ON THE FOLLOWING INFORMATION: Let X represent the number of times a student visits a bookstore in a -month period. Assume that the probability distribution of X is as follows: p() Find the mean µ and the standard deviation σ of this distribution. µ =.85, and σ = Find the mean and the standard deviation of Y = X. µy =.70, and σy =.584

12 Chapter Seven What is the probability that the student visits the bookstore at least once in a month? What is the probability that the student visits the bookstore at most twice in a month? 0.80 QUESTIONS 7 THROUGH 77 ARE BASED ON THE FOLLOWING INFORMATION: Let X and Y be two independent random variables with the following probability distributions: p() y p(y) Calculate E(X) and E(Y) E(X) = 0.70, E(Y) = Calculate V(X) and V(Y) V(X) = 0.6, V(Y) = Find the probability distribution of the random variable X + Y. +y p(+y) Calculate E(X + Y) directly by using the probability distribution of X + Y, and verify that E(X + Y) = E(X) + E(Y). E(X + Y) =.40. E(X) + E(Y) = =.40 = E(X + Y) 75. Calculate V(X + Y) directly by using the probability distribution of X + Y, and verify that V(X + Y) = V(X) + V(Y) for the independent variables X and Y.

13 Random Variables and Discrete Probability Distributions 06 V(X + Y) =.0. V(X) + V(Y) = =.0 = V(X + Y) 76. Find the probability distribution of the random variable XY. y p(y) Calculate E(XY) directly by using the probability distribution of XY, and verify that E(XY) = E(X).E(Y). E(XY) = 0.49 E(X).E(Y) = (0.70)(0.70) = 0.49 = E(XY). QUESTIONS 78 THROUGH 87 ARE BASED ON THE FOLLOWING INFORMATION: The joint probability distribution of variables X and Y is shown in the table below. Y 78. X Calculate E(XY) Determine the marginal probability distributions of X and Y. 80. p() y p(y) Calculate E(X) and E(Y)

14 07 Chapter Seven E(X) =.70, and E(Y) = Calculate V(X) and V(Y) ANSWERS V(X) = 0.6, and V(Y) = Are X and Y independent? Eplain. Yes, because p(,y) = p().p(y) for all pairs (,y). 8. Find P(Y = X = ) Calculate COV(X,Y). Did you epect this answer? Why? COV(X,Y) = E(XY) E(X).E(Y) =.55 (.70)(.50) = 0.0. Yes, since X and Y are independent per Question Find the probability distribution of the random variable X + Y y p(+y) Calculate E(X + Y) directly by using the probability distribution of X + Y. E(X + Y) = Calculate V(X + Y) directly by using the probability distribution of X + Y, and verify that V(X + Y) = V(X) + V(Y). Did you epect this result? Why? V(X + Y) =.06 V(X) + V(Y) = =.06 = V(X + Y) Yes, since X and Y are independent random variables. QUESTIONS 88 THROUGH 9 ARE BASED ON THE FOLLOWING INFORMATION: The joint probability distribution of X and Y is shown in the accompanying table. X Y

15 Random Variables and Discrete Probability Distributions Calculate E(XY) E(XY) = Determine the marginal probability distributions of X and Y. 90. p() y p(y) Find the mean, variance, and standard deviation for X and for Y. E(X) =., V(X) = 0., σ = E(Y) =.5, V(Y) = 0.5, σy = Calculate the conditional probabilities. P(X = Y = P(X = Y = P(X = Y = P(X = Y = 9. ) ) ) ) = = = = ,,,, P(Y = X = P(Y = X = P(Y = P(Y = X = Are X and Y independent? Eplain. No, because P(X = ) = ) = 0.57 ) = 0. X = ) = 0.49 ) = P(X = Y = ) = 0.8 Compute the covariance and the coefficient of correlation. COV(X,Y) = E(XY) E(X).E(Y) =.0 (.0)(.50) = 0.05, and ρy = COV(X,Y) / σ σy = 0.05 / 0.9 = Evaluate the following binomial coefficients. a. C 6 08

16 09 Chapter Seven 6 b. C c. C 7 d. C 47 a. 5 b. 0 c. d. 5 QUESTIONS 95 AND 96 ARE BASED ON THE FOLLOWING INFORMATION: An analysis of the stock market produces the following information about the returns of two stocks. Epected Returns Standard Deviations Stock 5% 0 Stock 8% Assume that the returns are positively correlated with ρ = Find the mean and standard deviation of the return on a portfolio consisting of an equal investment in each of the two stocks. The epected return of the portfolio: E ( R p ) = w E ( R ) + w E ( R ) = 0.5 (0.5) + 0.5(0.8) = 0.65 or 6.5%. The variance of the portfolio s return: V ( R p ) = wσ + wσ + ww ρσ σ = (0.5) (0) + (0.5) () + (0.5)(0.5)(0.8)(0)() = 6 Therefore, the standard deviation of the portfolio s return is 4.74% 96. Suppose that you wish to invest $ million. Discuss whether you should invest your money in stock, stock, or a portfolio composed of an equal amount of investments on both stocks. Your choice of investment in stock, the portfolio, or stock, depends on your desired level of risk (variance of return). The higher the risk you choose, the higher will be the epected return.

17 Random Variables and Discrete Probability Distributions Consider a binomial random variable X with n = 5 and p = a. Find the probability distribution of X. b. Find P(X < ). c. Find P( X 4). d. Find the mean and the variance of X. a. 0 p() b c d. E(X) =, and V(X) =. 98. Compute the following Poisson probabilities (to 4 decimal places) using the formula: a. P(X = ), if µ =.5 b. P(X ), if µ =.0 c. P(X ), if µ =.0 a. 0.8 b c Given a binomial random variable with n = 0 and p = 0.60, find the following probabilities using the binomial table. a. P(X ) b. P(X 5) c. P(X = 7) d. P( X 4) e. P( < X < 4) a b. 0.6 c. 0.0 d e A recent survey in Michigan revealed that 60% of the vehicles traveling on highways, where speed limits are posted at 70 miles per hour, were eceeding the limit. Suppose you randomly record the speeds of ten vehicles traveling on US where the speed limit is 70 miles per hour. Let X denote the number of vehicles that were eceeding the limit. Find the following probabilities: a. P(X = 0) b. P(4 < X < 9)

18 Chapter Seven c. P(X = ) d. P( X 6) X is a binomial random variable with n = 0 and p = a b c. 0.0 d An official from the securities commission estimates that 75% of all investment bankers have profited from the use of insider information. If 5 investment bankers are selected at random from the commission s registry, find the probability that: a. at most 0 have profited from insider information. b. at least 6 have profited from insider information. c. all 5 have profited from insider information. a. 0.4 b c A market researcher selects 0 students at random to participate in a winetasting test. Each student is blindfolded and asked to take a drink out of each of two glasses, one containing an epensive wine and the other containing a cheap wine. The students are then asked to identify the more epensive wine. If the students have no ability whatsoever to discern the more epensive wine, what is the probability that the more epensive wine will be correctly identified by: a. more than half of the students? b. none of the students? c. all of the students? d. Eight of the students? Let X denote the number of students who correctly identify the more epensive wine. X is a binomial random variable with n = 0 and p = 0.5. a. 0.4 b. 0.0 c. 0.0 d Let X be a Poisson random variable with µ = 6. Use the table of Poisson probabilities to find:

19 Random Variables and Discrete Probability Distributions a. b. c. d. P(X 8) P(X = 8) P(X 5) P(6 X 0) a b. 0.0 c d Let X be a Poisson random variable with µ = 8. Use the table of Poisson probabilities to find: a. P(X 6) b. P(X = 4) c. P(X ) d. P(9 X 4) a. 0. b c d Suppose you make a $,000 investment in a risky venture. There is a 50% chance that the payoff from the investment will be $5,000, a 0% chance that you will just get your money back, and a 0% chance that you will receive nothing at all from your investment. a. Find the epected value of the payoff from your investment of $,000. b. Find the epected value of the net profit from your investment of $,000. c. If you invest $6,000 in the risky venture instead of $,000, and the possible payoffs triple accordingly, what are the epected value of the net profit from the $6,000 investment? a. Epected value of the payoff from the $,000 investment: E ( R p ) = w E ( R ) + w E ( R ) + w E ( R ) = (0.5)($5000) + (0.)($000) + (0.) ($0) = $900 b. The epected value of the net profit from the $,000 investment: E(net profit ) = $900 - $000 = $900 c. Epected value of the net profit from the $6,000 investment = ($900) = $, The proprietor of a small variety store employs three men and three women. He will select three employees at random to work on Christmas Eve. Let X represent the number of women selected. a. Epress the probability distribution of X in tabular form.

20 Chapter Seven b. What is the probability that at least two women will work on Christmas Eve? a. 0 p() b Phone calls arrive at the rate of 0 per hour at the reservation desk for a hotel. a. Find the probability of receiving two calls in a five-minute interval of time. b. Find the probability of receiving eactly eight calls in 5 minutes. c. If no calls are currently being processed, what is the probability that the desk employee can take four minutes break without being interrupted? a. µ = 5(0/60) =.5; P(X = ) = b. µ = 5(0/60) = 7.5; P(X = 8) = 0.7 c. µ = 4(0/60) =.0; P(X = 0) = An advertising eecutive receives an average of 0 telephone calls each afternoon between and 4 P.M. The calls occur randomly and independently of one another. a. Find the probability that the eecutive will receive calls between and 4 P.M. on a particular afternoon. b. Find the probability that the eecutive will receive seven calls between and P.M. on a particular afternoon. c. Find the probability that the eecutive will receive at least five calls between and 4 P.M. on a particular afternoon. a. µ = 0; P(X = ) = 0.07 b. µ = 5; P(X = 7) = 0.05 c. µ = 0; P(X 5) = The number of arrivals at a local gas station between :00 and 5:00 P.M. has a Poisson distribution with a mean of. a. Find the probability that the number of arrivals between :00 and 5:00 P.M. is at least 0.

21 Random Variables and Discrete Probability Distributions b. Find P.M. c. Find P.M. 4 the probability that the number of arrivals between :0 and 4:00 is at least 0. the probability that the number of arrivals between 4:00 and 5:00 is eactly two. a. µ =; P(X 0) = b. µ = ; P(X 0) = 0.00 c. µ = 6; P(X = ) = Let X be a binomial random variable with n = 5 and p = 0.0. a. Use the binomial table to find P(X = 0), P(X = ), and P(X = ) b. Find the variance and standard deviation of X a. P(X = 0) = 0.778, P(X = ) = 0.96, and P(X = ) = 0.04 b. σ = np( p ) = 5(0.0)(0.99) = 0.475, and σ = The monthly sales at a bookstore have a mean of $50,000 and a standard deviation of $6,000. Profits are calculated by multiplying sales by 40% and subtracting fied costs of $,000. Find the mean and standard deviation of monthly profits. Let P = profit, and X = sales. Then P = 0.40X,000. E(P) = E(0.40X,000) = 0.40 E(X),000 = 0.40($50,000) - $,000 = $8,000 V(P) = V(0.40X,000) = (0.40) V(X) = (0.40) (6,000) = 5,760,000. Thus, the standard deviation of monthly profits is σ =$,400 QUESTIONS THROUGH 6 ARE BASED ON THE FOLLOWING INFORMATION: An investor has decided to form a portfolio by putting 0% of her money into stock and 70% into stock. The investor assumes that the epected returns will be 0% and 8%, respectively, and that the standard deviations will be 5% and 4%, respectively.. Find the epected mean of the portfolio. The epected return of the portfolio: E ( R p ) = w E ( R ) + w E ( R ) = 0. (0.0) (0.8) = 0.56 or 5.6%.. Compute the standard deviation of the returns on the portfolio assuming that the two stocks returns are perfectly positively correlated.

22 5 Chapter Seven The variance of the portfolio s return: V ( R p ) = wσ + wσ + ww ρσ σ = (0.) (0.5) +(0.7) (0.4) + (0.)(0.7)( ρ )(0.5)(0.4) = ρ When ρ = ; V ( R p ) = Therefore, the standard deviation of the portfolio s return is 0. or.% 4. Compute the standard deviation of the returns on the portfolio assuming that the coefficient of correlation is 0.5 When ρ = 0.5; V ( R p ) = Therefore, the standard deviation of the portfolio s return is 0.94 or 9.4% 5. Compute the standard deviation of the returns on the portfolio assuming that the two stocks returns are uncorrelated. When ρ = 0.0; V ( R p ) = Therefore, the standard deviation of the portfolio s return is 0.74 or 7.4% 6. Describe what happens to the standard deviation of the portfolio returns when the coefficient of correlation decreases. The standard deviation of the portfolio returns decreases as the coefficient of correlation decreases.