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introduction to probability statistics

Introduction To Probability Models 12th Edition Sheldon M Ross - Solutions

Find the variance for each of the random variables in Exercise 1 of Section 4.4.
Give an example of two random variables X and Y such that neither of the expectations E(X) and E(Y) exists, but the expected value of the sum X + Y is finite.
An urn contains 6 red balls numbered 1, 2, 3, 4, 5, 6, and 4 black balls with numbers 7, 8, 9, 10.We select three balls from the urn randomly and without replacement. For each of the following variables, find the probability function and its expectation:(i) the number of red balls selected,(ii) the
A ballot contains N chips numbered 1, 2,…,N.We select k chips with replacement.Let X be the largest number on the chips drawn.(i) Find the probability function f (x) = P(X = x).(ii) Find the expected value of X.(Note that this is not the same setup as in Example 4.10, because there we had no
A box contains five cyclical discs, with diameters 1, 2, 3, 4, and 5 cm, respectively.We select a disc at random.(i) What is the expected length of the disc chosen?(ii) What is the expected area of the disc chosen?
The probability function of a discrete random variable X has the form f (x) = c(x2 + 2|x| + 1), x ∈ RX = {−2,−1, 0, 1, 2}, for a suitable constant c.(i) Find the value of c.(ii) Calculate E(X) and then find the expectation of the random variables Y = |X|and W = 5X − 3|X|.
A company wants to promote two new products, a andb. For this purpose, the company sends a salesman to visit a number of houses in order to find buyers for these products. If someone does not buy either of the two products, the company gives them a small gift that costs $10. If a person buys
Bill plays the following game: he flips a coin four times. If the number of times that heads appear is 0 or 1, he wins $6, if heads appear 3 or 4 times he loses $12.What is the amount of money he should receive if heads appear twice so that this is a fair game?
Nick throws a die and if the outcome is 1 or 6, he gives Peter $9, while if the outcome is 3 he gives him $6. What amount should Peter pay Nick if the outcome is either 2 or 4 so that they have a fair game? (If the outcome of the die is 5, there is no money exchange between the two players.)
A friend of yours is proposing the following game: you put nine balls numbered 1, 2,…, 9 in a box and you select one at random. If the number on the ball drawn is a multiple of 3, he will give you $100. How many dollars should you give him if the number of the ball is not divisible by 3 so that
An insurance company has estimated, using previous data that, during a year, an insured person has a probability p of making a claim worth x thousands of dollars(i.e. all claims are of a fixed amount), and probability 1 − p of making no claims.What is the annual premium the company should charge
In an army camp, a large number, N, of soldiers are subject to a blood test to examine whether they have a certain disease. There are two ways to carry out this experiment:A. each soldier is tested separately, so that N tests take place;B. the blood samples of k soldiers (k < N) can be pooled and
A small store buys every week six 2-l bottles of milk at a price of $1.25 each. If at the end of the week a bottle has not been sold, the store returns it to the provider and receives 30 cents back. The probabilities that the number of 2-l bottles sold during a week is 0, 1, 2, 3, 4, 5, and 6 equal
At a Christmas bazaar, 6000 lottery tickets were sold at a price of $5 each. There are 15 prizes to be won from the lottery: a car with a value of $15 000, three trips to exotic locations, each of which costs $2500, a TV set whose value is $700 and 10 mobile phones each of which costs $250. Helena
Suppose that a player bets on red or black. Half of the numbers from 1 to 36 are red, the rest are black. The player places a bet of $10 on red; if the ball stops on a red number, he receives his$10 back plus $10 more. If the ball stops on a black number, the 0 or the 00 slot, he loses his bet.
The roulette wheel in a casino has 38 slots. Two of the slots have a 0 and a 00, and the remaining slots have the integers from 1 to
A gambler pays €10 to enter the following game: he throws two dice and if the outcomes contain• exactly one ace, he receives €15;• two aces, he receives €30.If no ace appears, he receives nothing. Find the gambler’s expected profit from this game.
Jimmy participates in a TV quiz show in which he is given two multiple choice questions. In the first one, he has three possible answers and in the second there are four possible answers.(i) If he selects his answer to both questions completely at random, what is the expected number of correct
We toss three coins and suppose X denotes the number of heads that show up. Find the expected value of X.
The number of boats that arrive at a Greek island during a day is 0, 1, 2, 3, and 4 with probabilities 0.1, 0.3, 0.25, 0.15, and 0.2, respectively. What is the expected number of boats that arrive during a day?
The probability that a technician fixes on a given day 0, 1, 2, 3, 4, 5, and 6 electric appliances is 0.05, 0.10, 0.20, 0.25, 0.20, 0.10, and 0.10, respectively. What is the expected number of appliances that the technician fixes in a day?
In a TV quiz show, a contestant is given the names of three countries and three capital cities and she is asked to match every country with its capital. If the contestant makes the correspondence completely at random, find the probability function of the correct matches.
After a car accident, a hospitalized patient is in need of blood. Among five blood donors, only two have a blood type that matches the one needed for the patient.Find the probability function and the distribution function of the number of persons that will be examined until the first donor with the
From a box which contains n lottery tickets numbered 0, 1,…, n − 1, we select tickets with replacement until the number 0 appears. Let X be the number of selections made.(i) Find the probability function of X.(ii) What is the range and the probability function of the random variable 5X − 3?
In a population of microorganisms, we assume that each individual can produce 0, 1, or 2 new microorganisms with probabilities 1∕5, 3∕5, and 1∕5, respectively.Starting from a single individual (considered as the zeroth generation), let Xi be the number of microorganisms in the ith generation
Let f1 and f2 be the probability functions of two discrete random variables with the same range RX and 0 ≤ ???? ≤ 1 be a real number. Verify that the function f (x) = ????f1(x) + (1 − ????)f2(x), x ∈ RX, defines a probability function on RX. Observe that this is the probability function of
Sofia is taking three exams at the end of this semester, one in each of the following courses: Mathematics, Statistics, and Economics. She estimates that the probability of passing the Mathematics exam is 0.85, the Statistics exam 0.75, and the Economics exam 0.90. If the three events “Success in
What is the probability that he wins this bet?
From a box that contains 20 balls numbered 1–20, we select 3 balls without replacement. Let X be the largest number among the balls drawn.(i) Find the probability function of X.(ii) Nick bets that the largest number in the 3 balls drawn will be at least
We toss a coin four times in succession. Let X be the number of times that the ordered pair HT appears in this experiment. Find the probability function and the distribution function of X.
We are going to select at random a family of three children. Let X be the random variable that denotes the number of girls minus the number of boys in that family.Find the range of values for X and determine the probability function and the distribution function associated with it.
A random variable is said to be symmetric around zero if we have P(X ≥ t) = P(X ≤ −t)for any t ∈ ℝ. Prove that, if X is symmetric around zero with a distribution function F, then the following hold:(i) P(|X| ≤ t) = 2F(t) − 1;(ii) P(X = t) = F(t) + F(−t) − 1.If, in addition, the
In Example 4.7, you were given that the distribution F of the amount paid by an insurance company for a claim that arrives from one of its customers is given by(4.3) and that this amount lies in the range (0, 4).Without this last bit of information, would we able to determine c uniquely? Explain.
A company manager has two secretaries, Mary and Joanne. In each page typed by Mary, the number of misprints can be 0, 1, or 2 with probabilities 0.50, 0.35, and 0.15, respectively, while the number of misprints in a page typed by Joanne can also be 0, 1, or 2, but with probabilities 0.60, 0.30, and
Verify that the function F defined by F(t) = ????F1(t) + (1 − ????)F2(t), t ∈ ℝ, is also a distribution function. The distribution F(t) is referred to as a twocomponent mixture distribution.
Let F1 and F2 be the distribution functions of two random variables, and ???? be a real number such that 0 ≤ ???? ≤
From an ordinary deck of 52 cards, we select three cards at random. Let X be the number of aces drawn. Write down the distribution function of X. Is this a discrete or continuous distribution function?
With reference to Example 4.7, verify analytically (i.e. without the aid of Figure 4.3)that for c = 1∕16, the distribution function in (4.3) is an increasing function on the real line. At which values of t is F differentiable?
In a family with four children, let X be the number of girls. Write down the distribution function of X.
Identify the range of values for X in Example 4.6. Then calculate the probability P(1 < X < 2|X ≤ 2).
A bullet is shot against a target whose center is located at the point O with coordinates (0, 0) in a Cartesian system. The position of the point that the bullet hits the target is described by its coordinates (X, Y), which are random variables.The ranges of values for X and Y are RX = [−2, 2]
We select at random a point A in the interval [−1, 1].We then select a second point B, also at random, but now from the interval [−1, A]. Identify the sets representing the ranges of each of the following variables:(i) X: the position of the point A in the interval [−1, 1];(ii) Y: the
A gas station has four pumps with unleaded petrol and six pumps with diesel. For an arbitrary time instant during a day, find the range of values for each of the following variables:X: the number of pumps with unleaded petrol which are in use;Y: the total number of pumps in use;Z: the larger number
A building plot has the shape of a rectangle whose length and width are represented by two variables X and Y. The ranges of these two variables are the intervals RX = [5, 8] and RY = [4, 6]. Let E be the area of the building plot.(i) Find the range of values, RE, for this variable.(ii) Explain
When an oil tanker sinks, the radius of the contaminated area by the oil spot is represented by a random variable X.(i) What is the range of values for X?(ii) Explain how we can define a new variable Y that gives the area of the contaminated region in the sea. You may assume that the contaminated
Find the range of values for each of the following random variables:(i) the number of successful hits in n throws of a dart against a target;(ii) the age of a person randomly selected from a population;(iii) the number of throws of a coin until two successive outcomes are the same;(iv) the number
Nick throws a die four times in succession. Define the random variable X: the number of times that the outcome is heads.(i) Find the sample space Ω for this experiment.(ii) What is the range of values for X?(iii) Identify the elements that each of the following sets (events) contains:A1 = {????
14. With W and as defined in Section 12.7, show that(a)(b) If for each , W and can be coupled so that, show that
13. If X and Y are discrete integer valued random variables with respective mass functions and , show that
12. Let and be independent irreducible Markov chains with states , and with respective transition probabilities and .(a) Give the transition probabilities of the Markov chain.(b) Show by giving a counterexample that is not necessarily irreducible.
11. Let be a renewal process whose interarrival times, have distribution F.(a) The random variable is the length of the renewal interval that does what.(b) Show that .
10. If is a random variable whose distribution is that of the conditional distribution of X given that , show that for every a.
9. A discrete time birth and death process is a Markov chain with transition probabilities of the form . Prove or give a counterexample to the claim that is stochastically increasing in i.
8. Consider two renewal processes: and whose interarrival distributions are discrete with, respective, hazard rate functions and . For any set of points A, let and denote, respectively, the numbers of renewals that occur at time points in A for the two processes. If for all i and either or is
7. If X is a positive integer valued random variable, with mass function , , then the function is called the (discrete) hazard rate function of X.(a) Express in terms of the values .(b) If is increasing (decreasing) in i then the random variable X is said to have increasing (decreasing) failure
6. A new item will fail on its ith day of use with probability. An item that fails during a period is replaced by a new one at the beginning of the next period. Let denote the age of the item in use at the beginning of period n. That is, if the item in use is beginning its ith day. The random
5. Let , , be nonhomogeneous Poisson processes with respective intensity functions . Suppose for all t. Let be arbitrary subsets of the real line, and for , let be the number of points of the process that are in . Show that.
4. Let be a renewal process with interarrival distribution . If , show that .
3. Show that a gamma random variable, whose density is is stochastically increasing in n and stochastically decreasing in λ.
2. If , is it possible to have .
1. Show that a normal random variable is stochastically increasing in its mean. That is, with being a normal random variable with mean μ and variance , show that when.
17. Order Statistics: Let be i.i.d. from a continuous distribution F, and let denote the ith smallest of. Suppose we want to simulate. One approach is to simulate n values from F, and then order these values. However, this ordering, or sorting, can be time consuming when n is large.(a) Suppose that
*16. Suppose n balls having weightsI mage are in an urn. These balls are sequentially removed in the following manner: At each selection, a given ball in the urn is chosen with a probability equal to its weight divided by the sum of the weights of the other balls that are still in the urn. LetI
15. Suppose you have just simulated a normal random variable X with mean μ and variance . Give an easy way to generate a second normal variable with the same mean and variance that is negatively correlated with X.
14. The Discrete Hazard Rate Method: Let X denote a nonnegative integer valued random variable. The function, is called the discrete hazard rate function.(a) Show that .(b) Show that we can simulate X by generating random numbers stopping at(c) Apply this method to simulating a geometric random
13. The Discrete Rejection Method: Suppose we want to simulate X having probability mass functionI mage and suppose we can easily simulate from the probability mass functionI mage. Let C be such that . Show that the following algorithm generates the desired random variable:Step 1: Generate Y having
12. Let be independent with If D is the number of distinct values among show that
11. Complete the details of Example 11.10.
10. Explain how we can number the in the alias method so that k is one of the two points that gives weight.Hint: Rather than giving the initial Q the name , what else could we call it?
9. Set up the alias method for simulating from a binomial random variable with parameters .
8. Consider the technique of simulating a gamma random variable by using the rejection method with g being an exponential density with rateI mage.(a) Show that the average number of iterations of the algorithm needed to generate a gamma is Image.(b) Use Stirling's approximation to show that for
7. Give an algorithm for simulating a random variable having density function
*6. In Example 11.5 we simulated the absolute value of a standard normal by using the Von Neumann rejection procedure on exponential random variables with rate 1. This raises the question of whether we could obtain a more efficient algorithm by using a different exponential density—that is, we
5. Suppose it is relatively easy to simulate fromI mage for eachI mage.How can we simulate from(a) ?(b) ?(c) Give two methods for simulating from the distribution.
4. Suppose we want to simulate a point located at random in a circle of radius r centered at the origin. That is, we want to simulate having joint density(—that is, ifI mage—setI mage. Otherwise return to step 1.(b) Prove that this method works, and compute the distribution of the number of
4. Suppose we want to simulate a point located at random in a circle of radius r centered at the origin. That is, we want to simulate having joint density(Give a method for simulating from Image
*3. Give a method for simulating a hypergeometric random variable.
2. Give a method for simulating a negative binomial random variable.
*1. Suppose it is relatively easy to simulate from the distributions. If n is small, how can we simulate from Give a method for simulating from Image
37. Let be weakly stationary with covariance function and let denote the power spectral density of the process.(i) Show that . It can be shown that
36. Let and be independent unit normal random variables and for some constant w set(a) Show that is a weakly stationary process.(b) Argue that is a stationary process.
35. Let be a weakly stationary process having covariance function .(a) Show that
34. Let denote a Poisson process with rate λ and define to be the time from t until the next Poisson event.(a) Argue that is a stationary process.(b) Compute .
33. Let where is a Poisson process with rate λ. Compute
32. Let denote a Brownian bridge process. Show that if then is a standard Brownian motion process.
31. For , argue that and are independent.
30. Let for . Argue that is a standard Brownian motion process.
29. Let and .(a) What is the distribution of ?(b) Compare .(c) Argue that is a standard Brownian motion process.
28. Compute the mean and variance of(a)(b)
27. Determine the distribution function of .
26. Let be the first time that the process is equal to y. For , show that Let be the maximal value ever attained. Explain why the preceding implies that, when , M is an exponential random variable with rate .
25. Suppose every Δ time units a process either increases by the amount with probability p or decreases by the amount with probability where.Show that as Δ goes to 0, this process converges to a Brownian motion process with drift parameter μ and variance parameter .
*24. Let be Brownian motion with drift coefficient μ and variance parameter . Suppose that . Let and define the stopping time T (as in Exercise 21) by Use the Martingale defined in Exercise 18, along with the result of Exercise 21, to show that In Exercises 25 to 27, is a Brownian motion process
23. Let , and define T to be the first time the process hits either A or −B, where A and B are given positive numbers. Use the Martingale stopping theorem and part (c)of Exercise 22 to find .
22. Let , and for given positive constants A and B, let p denote the probability that hits A before it hits−B.(a) Define the stopping time T to be the first time the process hits either A or −B. Use this stopping time and the Martingale defined in Exercise 19 to show that(b) Let , and show
21. Let be Brownian motion with drift coefficient μ and variance parameter . That is,Let , and for a positive constant x let That is, T is the first time the process hits x. Use the Martingale stopping theorem to show that
*20. Let That is, T is the first time that standard Brownian motion hits the line . Use the Martingale stopping theorem to find .
*19. Show that is a Martingale when where c is an arbitrary constant. What is ?An important property of a Martingale is that if you continually observe the process and then stop at some time T, then, subject to some technical conditions (which will hold in the problems to be considered), The time T
18. Show that is a Martingale when What is ?Hint: First compute .
17. Show that standard Brownian motion is a Martingale.

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