15.3 Problems on Random Selection

(See Exercise 3 from "Problems on Random Variables and Joint Distributions") A die is rolled. Let X be the number of spots that turn up. A coin is flipped X times. Let Y be the number of heads that turn up. Determine the distribution for Y.

(See Exercise 4 from "Problems on Random Variables and Joint Distributions") As a variation of the exercise, suppose a pair of dice is rolled instead of a single die. Determine the distribution for Y.

(See Exercise 5 from "Problems on Random Variables and Joint Distributions") Suppose a pair of dice is rolled. Let X be the total number of spots which turn up. Roll the pair an additional X times. Let Y be the number of sevens that are thrown on the X rolls. Determine the distribution for Y. What is the probability of three or more sevens?

(See Example 7 from "Conditional Expectation, Regression") A number X is chosen by a random selection from the integers 1 through 20 (say by drawing a card from a box). A pair of dice is thrown X times. Let Y be the number of “matches” (i.e., both ones, both twos, etc.). Determine the distribution for Y.

(See Exercise 20 from "Problems on Conditional Expectation, Regression") A number X is selected randomly from the integers 1 through 100. A pair of dice is thrown X times. Let Y be the number of sevens thrown on the X tosses. Determine the distribution for Y. Determine $E \left[Y\right]$ and $P \left(Y \leq 20\right)$.

(See Exercise 21 from "Problems on Conditional Expectation, Regression") A number X is selected randomly from the integers 1 through 100. Each of two people draw X times independently and randomly a number from 1 to 10. Let Y be the number of matches (i.e., both draw ones, both draw twos, etc.). Determine the distribution for Y. Determine $E \left[Y\right]$ and $P \left(Y \leq 10\right)$.

Suppose the number of entries in a contest is $N \sim$ binomial (20, 0.4). There are four questions. Let Y_i be the number of questions answered correctly by the ith contestant. Suppose the Y_i are iid, with common distribution

Let D be the total number of correct answers. Determine $E \left[D\right] , Var \left[D\right]$, $P \left(15 \leq D \leq 25\right)$, and $P \left(10 \leq D \leq 30\right)$.

Game wardens are making an aerial survey of the number of deer in a park. The number of herds to be sighted is assumed to be a random variable $N \sim$ binomial (20, 0.5). Each herd is assumed to be from 1 to 10 in size, with probabilities

Let D be the number of deer sighted under this model. Determine $P \left(D \leq t\right)$ for $t = 25 , 50 , 75 , 100$
and $P \left(D \geq 90\right)$.

A supply house stocks seven popular items. The table below shows the values of the items and the probability of each being selected by a customer.

Suppose the purchases of customers are iid, and the number of customers in a day is binomial (10,0.5). Determine the distribution for the total demand D.

1. How many different possible values are there? What is the maximum possible total sales?
2. Determine…

A game is played as follows:

1. A wheel is spun, giving one of the integers 0 through 9 on an equally likely basis.
2. A single die is thrown the number of times indicated by the result of the spin of the wheel. The number of points made is the total of the numbers turned up on the sequence of throws of the die.
3. A player pays sixteen dollars to play; a dollar is returned for each point made.

Let Y…

Marvin calls on four customers. With probability $p_{1} = 0 . 6$ he makes a sale in each case. Geraldine calls on five customers, with probability $p_{2} = 0 . 5$ of a sale in each case. Customers who buy do so on an iid basis, and order an amount Y_i (in dollars) with common distribution:

Let D₁ be the total sales for Marvin and D₂ the total sales for Geraldine. Let $D = D_{1} + D_{2}$. Determine the distribution and mean and variance for…

A questionnaire is sent to twenty persons. The number who reply is a random number $N \sim$ binomial (20, 0.7). If each respondent has probability $p = 0 . 8$ of favoring a certain proposition, what is the probability of ten or more favorable replies? Of fifteen or more?

A random number N of students take a qualifying exam. A grade of 70 or more earns a pass. Suppose $N \sim$ binomial (20, 0.3). If each student has probability $p = 0 . 7$ of making 70 or more, what is the probability all will pass? Ten or more will pass?

Five hundred questionnaires are sent out. The probability of a reply is 0.6. The probability that a reply will be favorable is 0.75. What is the probability of at least 200, 225, 250 favorable replies?

Suppose the number of Japanese visitors to Florida in a week is $N 1 \sim$ Poisson (500) and the number of German visitors is $N 2 \sim$ Poisson (300). If 25 percent of the Japanese and 20 percent of the Germans visit Disney World, what is the distribution for the total number D of German and Japanese visitors to the park? Determine $P \left(D \geq k\right)$ for $k = 150 , 155 , \hdots , 245 , 250$.

A junction point in a network has two incoming lines and two outgoing lines. The number of incoming messages N₁ on line one in one hour is Poisson (50); on line 2 the number is $N_{2} \sim$ Poisson (45). On incoming line 1 the messages have probability $p_{1 a} = 0 . 33$ of leaving on outgoing line a and $1 - p_{1 a}$ of leaving on line b. The messages coming in on line 2 have probability $p_{2 a} = 0 . 47$ of leaving on line a. Under the usual independ…

A computer store sells Macintosh, HP, and various other IBM compatible personal computers. It has two major sources of customers:

1. Students and faculty from a nearby university
2. General customers for home and business computing. Suppose the following assumptions are reasonable for monthly purchases.

• The number of university buyers $N 1 \sim$ Poisson (30). The probabilities for Mac, HP, others are 0.4, 0.2,…

The number N of “hits” in a day on a Web site on the internet is Poisson (80). Suppose the probability is 0.10 that any hit results in a sale, is 0.30 that the result is a request for information, and is 0.60 that the inquirer just browses but does not identify an interest. What is the probability of 10 or more sales? What is the probability that the number of sales is at least half the number of…

The number N of orders sent to the shipping department of a mail order house is Poisson (700). Orders require one of seven kinds of boxes, which with packing costs have distribution

What is the probability the total cost of the $2.50 boxes is no greater than $475? What is the probability the cost of the $2.50 boxes is greater than the cost of the $3.00 boxes? What is the probability the cost of t…

One car in 5 in a certain community is a Volvo. If the number of cars passing a traffic check point in an hour is Poisson (130), what is the expected number of Volvos? What is the probability of at least 30 Volvos? What is the probability the number of Volvos is between 16 and 40 (inclusive)?

A service center on an interstate highway experiences customers in a one-hour period as follows:

• Northbound: Total vehicles: Poisson (200). Twenty percent are trucks.
• Southbound: Total vehicles: Poisson (180). Twenty five percent are trucks.

• Each truck has one or two persons, with respective probabilities 0.7 and 0.3.
• Each car has 1, 2, 3, 4, or 5 persons, with probabilities 0.3, 0.3, 0.2,…

The number N of customers in a shop in a given day is Poisson (120). Customers pay with cash or by MasterCard or Visa charge cards, with respective probabilties 0.25, 0.40, 0.35. Make the usual independence assumptions. Let N₁, N₂, N₃ be the numbers of cash sales, MasterCard charges, Visa card charges, respectively. Determine $P \left(N_{1} \geq 30\right)$, $P \left(N_{2} \geq 60\right)$, $P \left(N_{3} \geq 50\right)$, and $P \left(N_{2} > N_{3}\right)$.

A discount retail store has two outlets in Houston, with a common warehouse. Customer requests are phoned to the warehouse for pickup. Two items, a and b, are featured in a special sale. The number of orders in a day from store A is $N_{A} \sim$ Poisson (30); from store B, the nember of orders is $N_{B} \sim$ Poisson (40).

For store A, the probability an order for a is 0.3, and for b is 0.7.

For store B, the probabilit…

The number of bids on a job is a random variable $N \sim$ binomial (7, 0.6). Bids (in thousands of dollars) are iid with Y uniform on $\left[3 , 5\right]$. What is the probability of at least one bid of $3,500 or less? Note that “no bid” is not a bid of 0.

The number of customers during the noon hour at a bank teller's station is a random number N with distribution

The amounts they want to withdraw can be represented by an iid class having the common distribution $Y \sim$ exponential (0.01). Determine the probabilities that the maximum withdrawal is less than or equal to t for $t = 100 , 200 , 300 , 400 , 500$.

A job is put out for bids. Experience indicates the number N of bids is a random variable having values 0 through 8, with respective probabilities

The market is such that bids (in thousands of dollars) are iid, uniform [100, 200]. Determine the probability of at least one bid of $125,000 or less.

A property is offered for sale. Experience indicates the number N of bids is a random variable having values 0 through 10, with respective probabilities

The market is such that bids (in thousands of dollars) are iid, uniform [150, 200] Determine the probability of at least one bid of $180,000 or more.

A property is offered for sale. Experience indicates the number N of bids is a random variable having values 0 through 8, with respective probabilities

The market is such that bids (in thousands of dollars) are iid symmetric triangular on [150 250]. Determine the probability of at least one bid of $210,000 or more.

Suppose $N \sim$ binomial $\left(10 , 0 . 3\right)$ and the Y_i are iid, uniform on $\left[10 , 20\right]$. Let V be the minimum of the N values of the Y_i. Determine $P \left(V > t\right)$ for integer values from 10 to 20.

Suppose a teacher is equally likely to have 0, 1, 2, 3 or 4 students come in during office hours on a given day. If the lengths of the individual visits, in minutes, are iid exponential (0.1), what is the probability that no visit will last more than 20 minutes.

Twelve solid-state modules are installed in a control system. If the modules are not defective, they have practically unlimited life. However, with probability $p = 0 . 05$ any unit could have a defect which results in a lifetime (in hours) exponential (0.0025). Under the usual independence assumptions, what is the probability the unit does not fail because of a defective module in the first 500 hours after…

The number N of bids on a painting is binomial $\left(10 , 0 . 3\right)$. The bid amounts (in thousands of dollars) Y_i form an iid class, with common density function $f_{Y} \left(t\right) = 0 . 005 \left(37 - 2 t\right)$$2 \leq t \leq 10$. What is the probability that the maximum amount bid is greater than $5,000?

A computer store offers each customer who makes a purchase of $500 or more a free chance at a drawing for a prize. The probability of winning on a draw is 0.05. Suppose the times, in hours, between sales qualifying for a drawing is exponential (4). Under the usual independence assumptions, what is the expected time between a winning draw? What is the probability of three or more winners in a ten h…

Noise pulses arrrive on a data phone line according to an arrival process such that for each $t > 0$ the number N_t of arrivals in time interval $\left(0 , t\right]$, in hours, is Poisson $\left(7 t\right)$. The ith pulse has an “intensity” Y_i such that the class $\left\{Y_{i} : 1 \leq i\right\}$ is iid, with the common distribution function $F_{Y} \left(u\right) = 1 - e^{- 2 u^{2}}$ for $u \geq 0$. Determine the probability that in an eight-hour day the intensity will not exceed two.

The number N of noise bursts on a data transmission line in a period $\left(0 , t\right]$ is Poisson $\left(\mu t\right)$. The number of digit errors caused by the ith burst is Y_i, with the class $\left\{Y_{i} : 1 \leq i\right\}$ iid, $Y_{i} - 1 \sim$ geometric $\left(p\right)$. An error correcting system is capable or correcting five or fewer errors in any burst. Suppose $\mu = 12$ and $p = 0 . 35$. What is the probability of no uncorrected error in two hours of operation?