Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y : f ( x , y ) = { x e − x ( 1 + y ) x ≥ 0 and y ≥ 0 0 otherwise a. What is the probability that the lifetime X of the first component exceeds 3? b. What are the marginal pdf’s of X and Y ? Are the two lifetimes independent? Explain. c. What is the probability that the lifetime of at least one component exceeds 3?

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Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809
BuyFind

Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809

Solutions

Chapter 5.1, Problem 12E
Textbook Problem

Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y:

     f ( x , y ) = { x e x ( 1 + y ) x 0 and  y 0 0 otherwise

  1. a. What is the probability that the lifetime X of the first component exceeds 3?
  2. b. What are the marginal pdf’s of X and Y? Are the two lifetimes independent? Explain.
  3. c. What is the probability that the lifetime of at least one component exceeds 3?

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Chapter 5 Solutions

Probability and Statistics for Engineering and the Sciences
Ch. 5.1 - Two different professors have just submitted final...Ch. 5.1 - Two components of a minicomputer have the...Ch. 5.1 - You have two lightbulbs for a particular lamp. Let...Ch. 5.1 - Suppose that you have ten lightbulbs, that the...Ch. 5.1 - Consider a system consisting of three components...Ch. 5.1 - a. For f(x1, x2, x3) as given in Example 5.10,...Ch. 5.1 - An ecologist wishes to select a point inside a...Ch. 5.1 - Refer to Exercise 1 and answer the following...Ch. 5.1 - The joint pdf of pressures for right and left...Ch. 5.1 - Let X1, X2, X3, X4, X5, and X6 denote the numbers...Ch. 5.1 - Let X1, X2, and X3 be the lifetimes of components...Ch. 5.2 - An instructor has given a short quiz consisting of...Ch. 5.2 - The difference between the number of customers in...Ch. 5.2 - Six individuals, including A and B, take seats...Ch. 5.2 - A surveyor wishes to lay out a square region with...Ch. 5.2 - Consider a small ferry that can accommodate cars...Ch. 5.2 - Annie and Alvie have agreed to meet for lunch...Ch. 5.2 - Show that if X and Y are independent rvs, then...Ch. 5.2 - Compute the correlation coefficient for X and Y...Ch. 5.2 - a. Compute the covariance for X and Y in Exercise...Ch. 5.2 - a. Compute the covariance between X and Y in...Ch. 5.2 - Reconsider the minicomputer component lifetimes X...Ch. 5.2 - Use the result of Exercise 28 to show that when X...Ch. 5.2 - a. Recalling the definition of 2 for a single rv...Ch. 5.2 - a. Use the rules of expected value to show that...Ch. 5.2 - Show that if Y = aX + b (a 0), then Corr(X, Y)...Ch. 5.3 - A particular brand of dishwasher soap is sold in...Ch. 5.3 - There are two traffic lights on a commuters route...Ch. 5.3 - It is known that 80% of all brand A external hard...Ch. 5.3 - A box contains ten sealed envelopes numbered 1, ....Ch. 5.3 - Let X be the number of packages being mailed by a...Ch. 5.3 - A company maintains three offices in a certain...Ch. 5.3 - Suppose the amount of liquid dispensed by a...Ch. 5.4 - Youngs modulus is a quantitative measure of...Ch. 5.4 - Refer to Exercise 46. 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Calculate the covariance...Ch. 5.5 - Suppose your waiting time for a bus in the morning...Ch. 5.5 - Suppose that when the pH of a certain chemical...Ch. 5.5 - If two loads are applied to a cantilever beam as...Ch. 5.5 - One piece of PVC pipe is to be inserted inside...Ch. 5.5 - Two airplanes are flying in the same direction in...Ch. 5.5 - Three different roads feed into a particular...Ch. 5.5 - Consider a random sample of size n from a...Ch. 5.5 - In Exercise 66, the weight of the beam itself...Ch. 5.5 - I have three errands to take care of in the...Ch. 5.5 - Suppose the expected tensile strength of type-A...Ch. 5.5 - In an area having sandy soil, 50 small trees of a...Ch. 5 - A restaurant serves three fixed-price dinners...Ch. 5 - In cost estimation, the total cost of a project is...Ch. 5 - A health-food store stocks two different brands of...Ch. 5 - According to the article Reliability Evaluation of...Ch. 5 - Suppose that for a certain individual, calorie...Ch. 5 - The mean weight of luggage checked by a randomly...Ch. 5 - We have seen that if E(X1) = E(X2) = =E(Xn) = ,...Ch. 5 - Suppose the proportion of rural voters in a...Ch. 5 - Let denote the true pH of a chemical compound. A...Ch. 5 - If the amount of soft drink that I consume on any...Ch. 5 - Refer to Exercise 58, and suppose that the Xis are...Ch. 5 - A student has a class that is supposed to end at...Ch. 5 - Garbage trucks entering a particular...Ch. 5 - Each customer making a particular Internet...Ch. 5 - a. Use the general formula for the variance of a...Ch. 5 - Suppose a randomly chosen individuals verbal score...Ch. 5 - a. Let X1 have a chi-squared distribution with...Ch. 5 - a. Show that Cov(X, Y + Z) = Cov(X, Y) + Cov(X,...Ch. 5 - A rock specimen from a particular area is randomly...Ch. 5 - Let A denote the percentage of one constituent in...Ch. 5 - Let X1, . . . , Xn be independent rvs with mean...Ch. 5 - A more accurate approximation to E[h(X1, . . . ,...Ch. 5 - Let X and Y be independent standard normal random...Ch. 5 - Let X1, X2, . . . , Xn be random variables...

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