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A First Course In Probability 9th Edition Sheldon Ross - Solutions

Solve the Banach match problem (Example 8e) when the left-hand matchbox originally contained N1 matches and the right-hand box contained N2 matches.
In the Banach matchbox problem, find the probability that, at the moment when the first box is emptied (as opposed to being found empty), the other box contains exactly k matches.
Suppose that a batch of 100 items contains 6 that are defective and 94 that are not defective. If X is the number of defective items in a randomly drawn sample of 10 items from the batch, find (a) P{X = 0} and (b) P{X > 2}.
There are three highways in the county. The number of daily accidents that occur on these highways are Poisson random variables with respective parameters .3, .5, and .7. Find the expected number of accidents that will happen on any of these highways today.
Suppose that 10 balls are put into 5 boxes, with each ball independently being put in box i with probability(a) Find the expected number of boxes that do not have any balls.(b) Find the expected number of boxes that have exactly 1 ball.
There are k types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of type i with probabilitythe expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set
There are N distinct types of coupons, and each time one is obtained it will, independently of past choices, be of type i with probability Pi, i = 1, . . . ,N. Let T denote the number one need select to obtain at least one of each type. Compute P{T = n}.
Let X be a binomial random variable with parameters n and p. Show that
Consider n independent sequential trials, each of which is successful with probability p. If there is a total of k successes, show that each of the n!/[k!(n − k)!] possible arrangements of the k successes and n − k failures is equally likely.
There are n components lined up in a linear arrangement. Suppose that each component independently functions with probability p. What is the probability that no 2 neighboring components are both nonfunctional?
Let X be a binomial random variable with parameters (n, p). What value of p maximizes P{X = k}, k = 0, 1, . . . , n? This is an example of a statistical method used to estimate p when a binomial (n, p) random variable is observed to equal k. If we assume that n is known, then we estimate p by
A family has n children with probability αpn, n ≥ 1, where α ≤ (1 − p)/p. (a) What proportion of families has no children? (b) If each child is equally likely to be a boy or a girl (independently of each other), what proportion of families consists of k boys (and any number of girls)?
Let X be a Poisson random variable with parameter λ.(a) Show thatP{X is even} = 1/2[1 + e−2λ]by using the result of Theoretical Exercise 15 and the relationship between Poisson and binomial random variables.(b) Verify the formula in part (a) directly by making use of the expansion of e−λ +
Let X be a Poisson random variable with parameter λ. What value of λ maximizes P{X = k}, k ≥ 0?
Show that X is a Poisson random variable with parameter λ, thenE[Xn] = λE[(X + 1)n−1]Now use this result to compute E[X3].
Consider n coins, each of which independently comes up heads with probability p. Suppose that n is large and p is small, and let λ = np. Suppose that all n coins are tossed; if at least one comes up heads, the experiment ends; if not, we again toss all n coins, and so on. That is, we stop the
From a set of n randomly chosen people, let Eij denote the event that persons i and j have the same birthday. Assume that each person is equally likely to have any of the 365 days of the year as his or her birthday. Find(a) P(E3,4|E1,2);(b) P(E1,3|E1,2);(c) P(E2,3|E1,2 ˆ© E1,3).What can you
An urn contains 2n balls, of which 2 are numbered 1, 2 are numbered 2, . . . , and 2 are numbered n. Balls are successively withdrawn 2 at a time without replacement. Let T denote the first selection in which the balls withdrawn have the same number (and let it equal infinity if none of the pairs
Consider a random collection of n individuals. In approximating the probability that no 3 of these individuals share the same birthday, a better Poisson approximation than that obtained in the text (at least for values of n between 80 and 90) is obtained by letting Ei be the event that there are at
Here is another way to obtain a set of recursive equations for determining Pn, the probability that there is a string of k consecutive heads in a sequence of n flips of a fair coin that comes up heads with probability p:(a) Argue that, for k < n, there will be a string of k consecutive heads if
If X is a geometric random variable, show analytically that P{X = n + k|X > n} = P{X = k} Using the interpretation of a geometric random variable, give a verbal argument as to why the preceding equation is true.
For a hypergeometric random variable, determineP{X = k + 1}/P{X = k}
Balls numbered 1 through N are in an urn. Suppose that n, n ≤ N, of them are randomly selected without replacement. Let Y denote the largest number selected.(a) Find the probability mass function of Y.(b) Derive an expression for E[Y] and then use Fermat’s combinatorial identity
A jar contains m + n chips, numbered 1, 2, . . . , n + m. A set of size n is drawn. If we let X denote the number of chips drawn having numbers that exceed each of the numbers of those remaining, compute the probability mass function of X.
From a set of n elements, a nonempty subset is chosen at random in the sense that all of the nonempty subsets are equally likely to be selected. Let X denote the number of elements in the chosen subset. Using the identities given in Theoretical Exercise 12 of Chapter 1, show thatShow also that, for
An urn initially contains one red and one blue ball. At each stage, a ball is randomly chosen and then replaced along with another of the same color. Let X denote the selection number of the first chosen ball that is blue. For instance, if the first selection is red and the second blue, then X is
Suppose the possible values of X are {xi}, the possible values of Y are {yj}, and the possible values of X + Y are {zk}. Let Ak denote the set of all pairs of indices (i, j) such that xi + yj = zk; that is,Ak = {(i, j): xi + yj = zk}.(a) Argue that(b) Show that(c) Using the formula from part (b),
Let X be such that P{X = 1} = p = 1 − P{X = −1} Find c ≠ 1 such that E[cX] = 1.
Let X be a random variable having expected value μ and variance σ2. Find the expected value and variance ofY = X - μ / σ
Find Var(X) if P(X = a) = p = 1 − P(X = b)
Show how the derivation of the binomial probabilitiesleads to a proof of the binomial theoremwhen x and y are nonnegative.Let p = x / x + y.
Let X be a random variable with probability density function(a) What is the value of c?(b) What is the cumulative distribution function of X?
Trains headed for destination A arrive at the train station at 15-minute intervals starting at 7 A.M., whereas trains headed for destination B arrive at 15-minute intervals starting at 7:05 A.M.(a) If a certain passenger arrives at the station at a time uniformly distributed between 7 and 8 A.M.
A point is chosen at random on a line segment of length L. Interpret this statement, and find the probability that the ratio of the shorter to the longer segment is less than 1/4.
Let X be a uniform (0, 1) random variable. Compute E[Xn] by using Proposition 2.1, and then check the result by using the definition of expectation.
If X is a normal random variable with parameters μ = 10 and σ2 = 36, compute (a) P{X > 5}; (b) P{4 < X < 16}; (c) P{X < 8}; (d) P{X < 20}; (e) P{X > 16}.
The annual rainfall (in inches) in a certain region is normally distributed with μ = 40 and σ = 4. What is the probability that, starting with this year, it will take over 10 years before a year occurs having a rainfall of over 50 inches? What assumptions are you making?
Suppose that X is a normal random variable with mean 5. If P{X > 9} = .2, approximately what is Var(X)?
Let X be a normal random variable with mean 12 and variance 4. Find the value of c such that P{X > c} = .10.
A system consisting of one original unit plus a spare can function for a random amount of time X. If the density of X is given (in units of months) bywhat is the probability that the system functions for at least 5 months?
If 65 percent of the population of a large community is in favor of a proposed rise in school taxes, approximate the probability that a random sample of 100 people will contain(a) At least 50 who are in favor of the proposition;(b) Between 60 and 70 inclusive who are in favor;(c) Fewer than 75 in
Every day Jo practices her tennis serve by continually serving until she has had a total of 50 successful sent If each of her serves is, independently of previous ones, successful with probability .4, approximately what is the probability that she will need more than 100 serves to accomplish her
One thousand independent rolls of a fair die will be made. Compute an approximation to the probability that the number 6 will appear between 150 and 200 times inclusively. If the number 6 appears exactly 200 times, find the probability that the number 5 will appear less than 150 times.
The lifetimes of interactive computer chips produced by a certain semiconductor manufacturer are normally distributed with parameters μ = 1.4 × 106 hours and σ = 3 × 105 hours. What is the approximate probability that a batch of 100 chips will contain at least 20 whose lifetimes are less than
Each item produced by a certain manufacturer is, independently, of acceptable quality with probability .95. Approximate the probability that at most 10 of the next 150 items produced are unacceptable.
In 10,000 independent tosses of a coin, the coin landed on heads 5800 times. Is it reasonable to assume that the coin is not fair? Explain.
Twelve percent of the population is left handed. Approximate the probability that there are at least 20 left-handers in a school of 200 students. State your assumptions.
A model for the movement of a stock supposes that if the present price of the stock is s, then, after one period, it will be either us with probability p or ds with probability 1 − p. Assuming that successive movements are independent, approximate the probability that the stock’s price will be
An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will give a reading that is normally distributed with μ = 4 and σ2 = 4, whereas one taken from a randomly chosen point in the black region will have a
(a) A fire station is to be located along a road of length A, A < ∞. If fires occur at points uniformly chosen on (0, A), where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as tominimize E[|X − a|]when X is uniformly distributed
Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with parameter 1/20. Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability
Suppose that the life distribution of an item has the hazard rate function λ(t) = t3, t > 0. What is the probability that (a) The item survives to age 2? (b) The item’s lifetime is between .4 and 1.4? (c) A 1-year-old item will survive to age 2?
If X is uniformly distributed over (−1, 1), find (a) P{|X| > 1/2}; (b) the density function of the random variable |X|.
If Y is uniformly distributed over (0, 5), what is the probability that the roots of the equation 4x2 + 4xY + Y + 2 = 0 are both real?
The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by(a) Find P{X > 20}.(b) What is the cumulative distribution function of X?(c) What is the probability that, of 6 such types of devices, at least 3 will function for at least
A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density functionwhat must the capacity of the tank be so that the probability of the supply€™s being exhausted in a given week is .01?
Compute E[X] if X has a density function given by(a)(b)(c)
The density function of X is given byIf E[X] = 3/5, find a and b.
Consider Example 4b of Chapter 4, but now suppose that the seasonal demand is a continuous random variable having probability density function f. Show that the optimal amount to stock is the value s∗ that satisfiesF(s∗) = b / b + ℓwhere b is net profit per unit sale, ℓ is the net loss per
The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability density function is given bywhere b = m/2kT and k, T, and m denote, respectively, Boltzmann€™s constant, the absolute temperature of the gas, and the mass of the molecule. Evaluate a in terms of b.
Let f(x) denote the probability density function of a normal random variable with mean μ and variance σ2. Show that μ − σ and μ + σ are points of inflection of this function. That is, show thatf″(x) = 0 when x = μ − σ or x = μ + σ.
Let Z be a standard normal random variable Z, and let g be a differentiable function with derivative g′.(a) Show that E[g′(Z)] = E[Zg(Z)](b) Show that E[Zn+1] = nE[Zn−1](c) Find E[Z4].
Use the identity of Theoretical Exercise 5 to derive E[X2] when X is an exponential random variable with parameter Î».Theoretical Exercise 5Use the result that, for a nonnegative random variable Y,to show that, for a nonnegative random variable X,and make the change of variables t = xn.
The median of a continuous random variable having distribution function F is that value m such that F(m) = 1/2. That is, a random variable is just as likely to be larger than its median as it is to be smaller. Find the median of X if X is (a) Uniformly distributed over (a, b); (b) Normal with
If X has hazard rate function λX(t), compute the hazard rate function of aX where a is a positive constant.
If X is an exponential random variable with mean 1/λ, show thatE[Xk] = k! / λkk = 1, 2, . . .
Show that
Verify that Var(X) = α / λ2 when X is a gamma random variable with parameters α and λ.
Show thatMake the change of variables y = ˆš2x and then relate the resulting expression to the normal distribution.
Compute the hazard rate function of a gamma random variable with parameters (α, λ) and show it is increasing when α ≥ 1 and decreasing when α ≤ 1.
LetY = (X – ν/α)βShow that if X is a Weibull random variable with parameters ν, α, and β, then Y is an exponential random variable with parameter λ = 1 and vice versa.
If X is a beta random variable with parameters a and b, show that E[X] = a/a + b Var(X) = ab/(a + b)2(a + b + 1)
Let X have probability density fX. Find the probability density function of the random variable Y defined by Y = aX + b.
Find the probability density function of Y = eX when X is normally distributed with parameters μ and σ2. The random variable Y is said to have a lognormal distribution (since log Y has a normal distribution) with parameters μ and σ2.
Prove Corollary 2.1.
Use the result that, for a nonnegative random variable Y,to show that, for a nonnegative random variable X,and make the change of variables t = xn.
Let X be a random variable that takes on values between 0 and c. That is, P{0 ≤ X ≤ c} = 1.Show thatVar(X) ≤ c2/4One approach is to first argue thatE[X2] ≤ cE[X]and then use this inequality to show thatVar(X) ≤ c2[α(1 − α)] where α =E[X]/c
Show that Z is a standard normal random variable, then, for x > 0,(a) P{Z > x} = P{Z < −x};(b) P{|Z| > x} = 2P{Z > x};(c) P{|Z| < x} = 2P{Z < x} − 1.
The joint probability density function of X and Y is given by f (x, y) = e−(x+y) 0 ≤ x < ∞, 0 ≤ y < ∞ Find (a) P{X < Y} (b) P{X < a}.
An ambulance travels back and forth at a constant speed along a road of length L. At a certain moment of time, an accident occurs at a point uniformly distributed on the road. [That is, the distance of the point from one of the fixed ends of the road is uniformly distributed over (0, L).] Assuming
The random vector (X, Y) is said to be uniformly distributed over a region R in the plane if, for some constant c, its joint density is(a) Show that 1/c = area of region R.Suppose that (X, Y) is uniformly distributed over the square centered at (0, 0) and with sides of length 2.(b) Show that X and
Suppose that n points are independently chosen at random on the circumference of a circle, and we want the probability that they all lie in some semicircle. That is, we want the probability that there is a line passing through the center of the circle such that all the points are on one side of
Two points are selected randomly on a line of length L so as to be on opposite sides of the midpoint of the line. [In other words, the two points X and Y are independent random variables such that X is uniformly distributed over (0, L/2) and Y is uniformly distributed over (L/2, L).] Find the
Show that f (x, y) = 1/x, 0 < y < x < 1, is a joint density function. Assuming that f is the joint density function of X,Y, find(a) The marginal density of Y;(b) The marginal density of X;(c) E[X];(d) E[Y].
Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8 red balls. Let Xi equal 1 if the ith ball selected is white, and let it equal 0 otherwise. Give the joint probability mass function of(a) X1, X2;(b) X1, X2, X3.
The joint density of X and Y is given by(a) Are X and Y independent? If, instead, f (x, y) were given by(b) Would X and Y be independent?
Letf (x, y) = 24xy0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ x + y ≤ 1and let it equal 0 otherwise.(a) Show that f (x, y) is a joint probability density function.(b) Find E[X].(c) Find E[Y].
The joint density function of X and Y is(a) Are X and Y independent?(b) Find the density function of X.(c) Find P{X + Y
The random variables X and Y have joint density function f (x, y) = 12xy(1 − x) 0 < x < 1, 0 < y < 1 and equal to 0 otherwise. (a) Are X and Y independent? (b) Find E[X]. (c) Find E[Y]. (d) Find Var(X). (e) Find Var(Y).
Consider independent trials, each of which results in outcome i, i = 0, 1, . . . , k, with probability pi,Let N denote the number of trials needed to obtain an outcome that is not equal to 0, and let X be that outcome.(a) Find P{N = n}, n ‰¥ 1.(b) Find P{X = j}, j = 1, . . . , k.(c) Show that
Suppose that A, B, C, are independent random variables, each being uniformly distributed over (0, 1).(a) What is the joint cumulative distribution function of A, B, C?(b) What is the probability that all of the roots of the equation Ax2 + Bx + C = 0 are real?
If X1 and X2 are independent exponential random variables with respective parameters λ1 and λ2, find the distribution of Z = X1/X2. Also compute P{X1 < X2}.
The time that it takes to service a car is an exponential random variable with rate 1.(a) If A. J. brings his car in at time 0 and M. J. brings her car in at time t, what is the probability that M. J.’s car is ready before A. J.’s car? (Assume that service times are independent and service
The gross weekly sales at a certain restaurant is a normal random variable with mean $2200 and standard deviation $230. What is the probability that(a) The total gross sales over the next 2 weeks exceeds $5000;(b) Weekly sales exceed $2000 in at least 2 of the next 3 weeks?What independence
In Problem 2, suppose that the white balls are numbered, and let Yi equal 1 if the ith white ball is selected and 0 otherwise. Find the joint probability mass function of(a) Y1, Y2;(b) Y1, Y2, Y3.Problem 2Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8
Jill’s bowling scores are approximately normally distributed with mean 170 and standard deviation 20, while Jack’s scores are approximately normally distributed with mean 160 and standard deviation 15. If Jack and Jill each bowl one game, then assuming that their scores are independent random
According to the U.S. National Center for Health Statistics, 25.2 percent of males and 23.6 percent of females never eat breakfast. Suppose that random samples of 200 men and 200 women are chosen. Approximate the probability that(a) At least 110 of these 400 people never eat breakfast;(b) The
The expected number of typographical errors on a page of a certain magazine is .2. What is the probability that an article of 10 pages contains(a) 0 and(b) 2 or more typographical errors?Explain your reasoning!
The monthly worldwide average number of airplane crashes of commercial airlines is 2.2. What is the probability that there will be (a) More than 2 such accidents in the next month? (b) More than 4 such accidents in the next 2 months? (c) More than 5 such accidents in the next 3 months? Explain your
In Problem 3, calculate the conditional probability mass function of Y1 given that(a) Y2 = 1;(b) Y2 = 0.Problem 3In Problem 2, suppose that the white balls are numbered, and let Yi equal 1 if the ith white ball is selected and 0 otherwise. Find the joint probability mass function ofProblem 2Suppose

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