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statistics for engineers and scientists

Introduction To Probability And Statistics For Engineers And Scientists 4th Edition Sheldon M. Ross - Solutions

Suppose that we are attempting to locate a target in three-dimensional space, and that the three coordinate errors (in meters) of the point chosen are independent normal random variables with mean 0 and standard deviation 2. Find the probability that the distance between the point chosen and the
A series system is one that needs all of its components to function in order for the system itself to be functional. For an n-component series system in which the component lifetimes are independent exponential random variables with respective parameters λ1, λ2, . . . , λn, what is the
A crew of workers has 3 interchangeable machines, of which 2 must be working for the crew to do its job. When in use, each machine will function for an exponentially distributed time having parameter λ before breaking down. The workers decide initially to use machines A and B and keep machine C in
Suppose that a number of miles that a car can run before its battery wears out is exponentially distributed with an average value of 10,000 miles. If a person desires to take a 5,000-mile trip, what is the probability that she will be able to complete her trip without having to replace her car
Data from theNationalOceanic and Atmospheric Administration indicate that the yearly precipitation in Los Angeles is a normal random variable with a mean of 12.08 inches and a standard deviation of 3.1 inches.(a) Find the probability that the total precipitation during the next 2 years will exceed
The power W dissipated in a resistor is proportional to the square of the voltage V. That is, W = rV 2 where r is a constant. If r = 3, and V can be assumed (to a very good approximation) to be a normal random variable with mean 6 and standard deviation 1, find(a) E [W];(b) P{W > 120}.
The current in a semiconductor diode is often measured by the Shockley equation I = I0(eaV − 1)where V is the voltage across the diode; I0 is the reverse current; a is a constant; and I is the resulting diode current. Find E[I] if a = 5, I0 = 10−6, and V is uniformly distributed over (1, 3).
Buses arrive at a specified stop at 15-minute intervals starting at 7 A.M. That is, they arrive at 7, 7:15, 7:30, 7:45, and so on. If a passenger arrives at the stop at a time that is uniformly distributed between 7 and 7:30, find the probability that he waits(a) less than 5 minutes for a bus;(b)
If X is uniformly distributed over the interval [0, 10], compute the probability that (a) 2 < X < 9, (b) 1 < X < 4, (c) X < 5, (d) X > 6.
The components of a 6-component system are to be randomly chosen from a bin of 20 used components. The resulting system will be functional if at least 4 of its 6 components are in working condition. If 15 of the 20 components in the bin are in working condition, what is the probability that the
It has been established that the number of defective stereos produced daily at a certain plant is Poisson distributed with mean 4. Over a 2-day span, what is the probability that the number of defective stereos does not exceed 3?
If the average number of claims handled daily by an insurance company is 5, what proportion of days have less than 3 claims? What is the probability that there will be 4 claims in exactly 3 of the next 5 days? Assume that the number of claims on different days is independent.
Consider an experiment that consists of counting the number of α particles given off in a 1-second interval by 1 gram of radioactive material. If we know from past experience that, on the average, 3.2 such α-particles are given off, what is a good approximation to the probability that no more
Suppose the probability that an item produced by a certain machine will be defective is .1. Find the probability that a sample of 10 items will contain at most one defective item. Assume that the quality of successive items is independent.
Suppose that the average number of accidents occurring weekly on a particular stretch of a highway equals 3. Calculate the probability that there is at least one accident this week.
If X is a binomial random variable with parameters n = 100 and p = .75, find P{X = 70} and P{X ≤ 70}.
Suppose that 10 percent of the chips produced by a computer hardware manufacturer are defective. If we order 100 such chips, will X, the number of defective ones we receive, be a binomial random variable?
A communications system consists of n components, each of which will, independently, function with probability p. The total system will be able to operate effectively if at least one-half of its components function.(a) For what values of p is a 5-component system more likely to operate effectively
The color of one’s eyes is determined by a single pair of genes, with the gene for brown eyes being dominant over the one for blue eyes. This means that an individual having two blue-eyed genes will have blue eyes, while one having either two brown-eyed genes or one brown-eyed and one blue-eyed
It is known that disks produced by a certain company will be defective with probability .01 independently of each other. The company sells the disks in packages of 10 and offers a money-back guarantee that at most 1 of the 10 disks is defective.What proportion of packages is returned? If someone
Let X and Y have respective distribution functions FX and FY , and suppose that for some constants a and b > 0,(a) Determine E[X ] in terms of E[Y ].(b) Determine Var(X ) in terms of Var(Y ).Hint: X has the same distribution as what other random variable? -a Fx(x) = Fy (x-4) b
From past experience, a professor knows that the test score of a student taking her final examination is a random variable with mean 75.(a) Give an upper bound to the probability that a student’s test score will exceed 85.Suppose in addition the professor knows that the variance of a student’s
Suppose that X is a random variable with mean and variance both equal to 20.What can be said about P{0 ≤ X ≤ 40}?
If the density function of X is f (x) = 1, 0 < x < 1 determine E[etX ]. Differentiate to obtain E[X n] and then check your answer.
Suppose that X has density function f (x) = e−x , x > 0 Compute the moment generating function of X and use your result to determine its mean and variance. Check your answer for the mean by a direct calculation.
If X1 and X2 have the same probability distribution function, show that Cov(X1 − X2, X1 + X2) = 0 Note that independence is not being assumed.
In Example 4.5f, compute Cov(Xi , Xj ) and use this result to show that Var(X ) = 1.
Consider n independent trials, each of which results in any of the outcomes i, i =1, 2, 3, with respective probabilities p1, p2, p3,3i=1 pi = 1. Let Ni denote the number of trials that result in outcome i, and show that Cov(N1,N2) = −np1p2.Also explain why it is intuitive that this covariance is
Let X have variance σ2 x and let Y have variance σ2 y . Starting withUsing the result that Var(Z) = 0 implies that Z is constant, argue that, if Corr(X , Y ) = 1 or −1, then X and Y are related by Y = a + bx where the sign of b is positive when the correlation is 1 and negative when it is −1.
Prove Equation 4.7.5 by using mathematical induction.
Verify Equation 4.7.4.
A machine makes a product that is screened (inspected 100 percent) before being shipped. The measuring instrument is such that it is difficult to read between 1 and 1*1/3(coded data). After the screening process takes place, the measured dimension has density(a) Find the value of k.(b) What
A product is classified according to the number of defects it contains and the factory that produces it. Let X1 and X2 be the random variables that represent the number of defects per unit (taking on possible values of 0, 1, 2, or 3) and the factory number (taking on possible values 1 or 2),
Suppose that the Rockwell hardness X and abrasion loss Y of a specimen (coded data) have a joint density given by(a) Find the marginal densities of X and Y .(b) Find E(X ) and Var(X ). u+v for 0 u, v 1 otherwise fxy (u,v) = {" 0
A random variable X , which represents the weight (in ounces) of an article, has density function given by f (z),(a) Calculate the mean and variance of the random variable X .(b) The manufacturer sells the article for a fixed price of $2.00. He guarantees to refund the purchase money to any
Argue that for any random variable XWhen does one have equality? E[X] (E[X])
Compute the mean and variance of the number of heads that appear in 3 flips of a fair coin.
Let pi = P{X = i} and suppose that p1 +p2 +p3 = 1. If E[X] = 2, what values of p1, p2, p3 (a) maximize and (b) minimize Var(X )
Suppose that X is equally likely to take on any of the values 1, 2, 3, 4. Compute(a) E[X ] and (b) Var(X ).
Compute the expectation and variance of the number of successes in n independent trials, each of which results in a success with probability p. Is independence necessary?
A community consists of 100 married couples. If during a given year 50 of the members of the community die, what is the expected number of marriages that remain intact? Assume that the set of people who die is equally likely to be any of the%200 50&groups of size 50. (Hint: For i = 1, . . . ,
We say that mp is the 100p percentile of the distribution function F if F (mp) = p Find mp for the distribution having density function f (x) = 2e−2x , x ≥ 0
The median, like the mean, is important in predicting the value of a random variable. Whereas it was shown in the text that the mean of a random variable is the best predictor from the point of view of minimizing the expected value of the square of the error, the median is the best predictor if one
If X is a continuous random variable having distribution function F , then its median is defined as that value of m for which F (m) = 1/2 Find the median of the random variables with density function(a) f (x) = e−x , x ≥ 0;(b) f (x) = 1, 0 ≤ x ≤
Ten balls are randomly chosen from an urn containing 17 white and 23 black balls.Let X denote the number of white balls chosen. Compute E[X ](a) by defining appropriate indicator variables Xi , i = 1, . . . , 10 so that(b) by defining appropriate indicator variables Yi = 1, . . . , 17 so that 10 X
If E[X] = 2 and E[X 2] = 8, calculate (a) E[(2+4X )2] and (b) E[X 2+(X +1)2].
The time it takes to repair a personal computer is a random variable whose density, in hours, is given byThe cost of the repair depends on the time it takes and is equal to 40 + 30 √x when the time is x. Compute the expected cost to repair a personal computer. f(x) = 0 < x < 2 0 otherwise
Suppose that X has density functionCompute E[X n] (a) by computing the density of Xn and then using the definition of expectation and (b) by using Proposition 4.5.1. 1 0 < x < 1 f(x) = 0 otherwise
Let X1, X2, . . . , Xn be independent random variables having the common density functionFind (a) E[Max(Xi , . . . , Xn)] and (b) E[Min(X1, . . . , Xn)]. 1 0 < x
The lifetime in hours of electronic tubes is a random variable having a probability density function given by f (x) = a2xe−ax , x ≥ 0 Compute the expected lifetime of such a tube.
The density function of X is given byIf E[X] = 35 , finda, b. f(x) a + bx 0 x 1 0 otherwise
Suppose that two teams play a series of games that end when one of them has won i games. Suppose that each game played is, independently, won by team A with probability p. Find the expected number of games that are played when i = 2.Also show that this number is maximized when p = 12.
A total of 4 buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying this randomly selected student. One of
An insurance company writes a policy to the effect that an amount of money A must be paid if some event E occurs within a year. If the company estimates that E will occur within a year with probability p, what should it charge the customer so that its expected profit will be 10 percent of A?
Each night different meteorologists give us the “probability” that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p, then he or she will receive a score of 1 − (1 − p)2 if
Compute the expected value of the random variable in Problem 3.
Compute the expected value of the random variable in Problem 1.
Show that X and Y are independent if and only if(a) pX |Y(x|y) = pX (x) in the discrete case(b) fX |Y(x|y) = fX (x) in the continuous case
Compute the conditional density function of X given Y = y in (a) Problem 10 and (b) Problem 13.
In Example 4.3b, determine the conditional probability mass function of the size of a randomly chosen family containing 2 girls.
When a current I (measured in amperes) flows through a resistance R (measured in ohms), the power generated (measured in watts) is given byW = I 2R. Suppose that I and R are independent random variables with densities fI (x) = 6x(1 − x) 0 ≤ x ≤ 1 fR(x) = 2x 0 ≤ x ≤ 1 Determine the density
Suppose that X and Y are independent continuous random variables. Show thatwhere fY is the density function of Y , and FX is the distribution function of X . (a) P{X + Y a} = = Fx (a - y)fy (y) dy (b) P(X Y) = Fx (y)fy(y) dy 00
Is Problem 14 consistent with the results of Problems 12 and 13?
If the joint density function of X and Y factors into one part depending only on x and one depending only on y, show that X and Y are independent.That is, if f (x, y) = k(x)l ( y), −∞ < x < ∞, −∞ < y < ∞show that X and Y are independent.
The joint density of X and Y is(a) Compute the density of X .(b) Compute the density of Y .(c) Are X and Y independent? 2 0 < x
The joint density of X and Y is given by(a) Compute the density of X .(b) Compute the density of Y .(c) Are X and Y independent? f(x,y) = Sxe(-x+y) x > 0, y >0 0 otherwise
Let X1, X2, . . . , Xn be independent random variables, each having a uniform distribution over (0, 1). LetM =maximum (X1, X2, . . . , Xn). Showthat the distribution function of M, FM(·), is given byWhat is the probability density function of M? FM(x)=x", 0 x 1
The joint probability density function of X and Y is given by(a) Verify that this is indeed a joint density function.(b) Compute the density function of X .(c) Find P{X > Y }. f(x, y) = 6 0
A bin of 5 transistors is known to contain 3 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N1 the number of tests made until the first defective is spotted and by N2 the number of additional tests until the second defective
If the density function of X equalsfindc. What is P{X > 2}? f(x) = = { ce -2x x < 0
The lifetime in hours of a certain kind of radio tube is a random variable having a probability density function given byWhat is the probability that exactly 2 of 5 such tubes in a radio set will have to be replaced within the first 150 hours of operation? Assume that the events Ei , i =1, 2, 3, 4,
The amount of time, in hours, that a computer functions before breaking down is a continuous random variable with probability density function given byWhat is the probability that a computer will function between 50 and 150 hours before breaking down? What is the probability that it will function
Suppose you are given the distribution function F of a random variable X . Explain how you could determine P{X = 1}. (Hint: You will need to use the concept of a limit.)
The distribution function of the random variable X is given(a) Plot this distribution function.(b) What is P{X > 12 }?(c) What is P{2 (d) What is P{X (e) What is P{X = 1}? 0 x
In Problem 2, if the coin is assumed fair, for n = 3, what are the probabilities associated with the values that X can take on?
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of X ?
Five men and 5 women are ranked according to their scores on an examination.Assume that no two scores are alike and all 10! possible rankings are equally likely.Let X denote the highest ranking achieved by a woman (for instance, X = 2 if the top-ranked person was male and the next-ranked person was
Suppose that it is known that the number of items produced in a factory during a week is a random variable with mean 50.(a) What can be said about the probability that this week’s production will exceed 75?(b) If the variance of a week’s production is known to equal 25, then what can be said
Compute the variance of the sum obtained when 10 independent rolls of a fair die are made.
Suppose there are 20 different types of coupons and suppose that each time one obtains a coupon it is equally likely to be any one of the types. Compute the expected number of different types that are contained in a set for 10 coupons
A secretary has typed N letters along with their respective envelopes. The envelopes get mixed up when they fall on the floor. If the letters are placed in the mixed-up envelopes in a completely random manner (that is, each letter is equally likely to end up in any of the envelopes), what is the
A construction firm has recently sent in bids for 3 jobs worth (in profits)10, 20, and 40 (thousand) dollars. If its probabilities of winning the jobs are respectively.2, .8, and .3, what is the firm’s expected total profit?
The time, in hours, it takes to locate and repair an electrical breakdown in a certain factory is a random variable — call it X — whose density function is given byIf the cost involved in a breakdown of duration x is x3, what is the expected cost of such a breakdown? fx (x) = = 1 if 0 < x
Suppose X has the following probability mass function p(0) = .2, p(1) = .5, p(2) = .3 Calculate E[X 2].
The joint density of X and Y is given byCompute the conditional density of X , given that Y = y, where 0 x(2-x-y) 0 < x < 1,0 < y < 1 f(x,y) = 10 otherwise
Suppose that p(x, y), the joint probability mass function of X and Y , is given by p(0, 0) = .4, p(0, 1) = .2, p(1, 0) = .1, p(1, 1) = .3 Calculate the conditional probability mass function of X given that Y = 1.
we know, in Example 4.3b, that the family chosen has one girl, compute the conditional probability mass function of the number of boys in the family.
The joint density function of X and Y is given byCompute (a) P{X > 1, Y (b) P{X (c) P{X 2ee2 0
Suppose that X is a continuous random variable whose probability density function is given by(a) What is the value of C?(b) Find P{X > 1}. f(x): C(4x-2x) 0
Two percent of woman of age 45 who participate in routine screening have breast cancer. Ninety percent of those with breast cancer have positive mammographies.Ten percent of the women who do not have breast cancer will also have positive mammographies. Given a woman has a positive mammography, what
Let A, B, C be events such that P(A) = .2, P(B) = .3, P(C) = .4.Find the probability that at least one of the events A and B occurs if(a) A and B are mutually exclusive;(b) A and B are independent.Find the probability that all of the events A, B, C occur if(c) A, B, C are independent;(d) A, B, C
Suppose that distinct integer values are written on each of 3 cards. Suppose you are to be offered these cards in a random order. When you are offered a card you must immediately either accept it or reject it. If you accept a card, the process ends.If you reject a card, then the next card (if a
A set of k coupons, each of which is independently a type j coupon with probability pj ,nj=1 pj =1, is collected. Find the probability that the set contains either a type i or a type j coupon.
Although both my parents have brown eyes, I have blue eyes. What is the probability that my sister has blue eyes?
Three prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this
A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase and capital letters). The capital letter will be assumed to be the dominant gene in the sense that
A parallel system functions whenever at least one of its components works. Consider a parallel system of n components, and suppose that each component independently works with probability 12. Find the conditional probability that component 1 works, given that the system is functioning.
Suppose that n independent trials, each of which results in any of the outcomes 0, 1, or 2, with respective probabilities .3, .5, and .2, are performed. Find the probability that both outcome 1 and outcome 2 occur at least once. (Hint: Consider the complementary probability.)
Five independent flips of a fair coin are made. Find the probability that(a) the first three flips are the same;(b) either the first three flips are the same, or the last three flips are the same;(c) there are at least two heads among the first three flips, and at least two tails among the last
An engineering system consisting of n components is said to be a k-out-of-n system (k ≤ n) if the system functions if and only if at least k of the n components function. Suppose that all components function independently of each other.

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