Probability and Statistics for Engineering and the Sciences
9th Edition
ISBN: 9781305251809
Author: Jay L. Devore
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Question
Chapter 5.1, Problem 15E
a.
To determine
Find the
b.
To determine
Find the expected system lifetime.
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
Let X ~ Beta (2, 2) and Y = 1 / X^2.
• Find the cumulative distribution function of Y.
• Calculate the expected excess loss function of Y.
• Determine whether the distribution of Y is light-tailed or heavy-tailed.
X1 and X2 are two discrete random variables, while the X1 random variable takes the values x1 = 1, x1 = 2 and x1 = 3, while the X2 random variable takes the values x2 = 10, x2 = 20 and x2 = 30. The combined probability mass function of the random variables X1 and X2 (pX1, X2 (x1, x2)) is given in the table below
a) Find the marginal probability mass function (pX1 (X1)) of the random variable X1.b) Find the marginal probability mass function (pX2 (X2)) of the random variable X2.c) Find the expected value of the random variable X1.d) Find the expected value of the random variable X2.e) Find the variance of the random variable X1.f) Find the variance of the random variable X2.g) pX1 | X2 (x1 | x2 = 10) Find the mass function of the given conditional probability.h) pX2 | X1 (x2 | x1 = 2) Find the mass function of the given conditional probability.i) Are the random variables X1 and X2 independent? Show it.
The combined probability mass function of the random variables X1 and X2 is below
LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0,
fXi(x) ={λe−λx, x >0
0, otherwise
Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.
Chapter 5 Solutions
Probability and Statistics for Engineering and the Sciences
Ch. 5.1 - Prob. 1ECh. 5.1 - A large but sparsely populated county has two...Ch. 5.1 - A certain market has both an express checkout line...Ch. 5.1 - Return to the situation described in Exercise 3....Ch. 5.1 - The number of customers waiting for gift-wrap...Ch. 5.1 - Let X denote the number of Canon SLR cameras sold...Ch. 5.1 - Prob. 7ECh. 5.1 - A stockroom currently has 30 components of a...Ch. 5.1 - Each front tire on a particular type of vehicle is...Ch. 5.1 - Prob. 10E
Ch. 5.1 - Two different professors have just submitted final...Ch. 5.1 - Two components of a minicomputer have the...Ch. 5.1 - Prob. 13ECh. 5.1 - Suppose that you have ten lightbulbs, that the...Ch. 5.1 - Prob. 15ECh. 5.1 - Prob. 16ECh. 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 - Prob. 26ECh. 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 - Prob. 30ECh. 5.2 - a. Compute the covariance between X and Y in...Ch. 5.2 - Reconsider the minicomputer component lifetimes X...Ch. 5.2 - Prob. 33ECh. 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. Suppose the distribution is...Ch. 5.4 - The National Health Statistics Reports dated Oct....Ch. 5.4 - There are 40 students in an elementary statistics...Ch. 5.4 - Let X denote the courtship time for a randomly...Ch. 5.4 - The time taken by a randomly selected applicant...Ch. 5.4 - The lifetime of a certain type of battery is...Ch. 5.4 - Rockwell hardness of pins of a certain type is...Ch. 5.4 - Suppose the sediment density (g/cm) of a randomly...Ch. 5.4 - The number of parking tickets issued in a certain...Ch. 5.4 - A binary communication channel transmits a...Ch. 5.4 - Suppose the distribution of the time X (in hours)...Ch. 5.5 - A shipping company handles containers in three...Ch. 5.5 - Let X1, X2, and X3 represent the times necessary...Ch. 5.5 - Refer back to Example 5.31. Two cars with...Ch. 5.5 - Exercise 26 introduced random variables X and Y,...Ch. 5.5 - Manufacture of a certain component requires three...Ch. 5.5 - Refer to Exercise 3. a. 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 - Prob. 77SECh. 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 - Prob. 91SECh. 5 - Prob. 92SECh. 5 - Prob. 93SECh. 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 - Prob. 97SECh. 5 - Prob. 98SE
Knowledge Booster
Similar questions
- Find X2 (the probability distribution of the system after two observations) for the distribution vector X0 and the transition matrix T. X0 = 0.9 0.1 , T = 0.1 0.6 0.9 0.4 X2 =arrow_forwardConsider a system consisting of three components as pictured. The system will continue to function as long as the first component functions and either component 2 or component 3 functions. Let X1, x2,and x3 denote the lifetimes of components 1,2,and 3,respectively.Suppose the Xi 'are independent of one another and each Xi has an exponential distribution with parameter λ. a. Let Y denote the system Obtain the cumulative distribution function of Yand differentiate to obtain the pdf.[ Hint: F(y) = P(Y≤S y);express the event {Y≤y} in terms of unions and/or intersections of the three events {X1 ≤ y}, {X2 ≤ y},and {x3 ≤ y}.] b. Compute the expected system lifetime.arrow_forwardFind Cumulative Distribution Function (cdf) of X, F(x), x {0, 1, 2,...}arrow_forward
- Fire load (MJ/m2) is the heat energy that could be released per square meter of floor area by combustion of contents and the structure itself. The following cumulative percentages are for fire loads in a sample of 388 rooms: Value 0 150 300 450 600 750 900 Cumulative % 0 19.0 37.3 62.0 77.0 86.7 93.4 Value 1050 1200 1350 1500 1650 1800 1950 Cumulative % 95.1 97.9 98.3 99.5 99.6 99.8 100.0 What proportion of fire loads are less than 600? At least 1200? (Round your answers to three decimal places.) less than 600 at least 1200 (c) What proportion of the loads are between 600 and 1200? (Round your answer to three decimal places.)arrow_forwardSample data on weekly sales of two dairy products, X and Y were reported as follows: Lilongwe dairy, X 120 100 160 250 180 130 300 40 Long life, Y 70 86 67 100 45 80 90 95 a. Apply the CV, and find out which of the 2 products shows greater fluctuations in sales.arrow_forwardGiven the Cumulative Distribution Function(C.D.F) F ( x ) = { 0 , x < − 2 (x + 3)/5 , − 2 ≤ x < 1.5 1 , 1.5 ≤ x Compute P(x>0)arrow_forward
- Suppose X has a trivariate normal distribution with mean vector 0 and covariance matrix 1 0.5 0.25 0.5 1 0 0.25 0 1 A. Find the joint distribution of W1 = X1+X2+X3 and W2 = X1 - X3 B. Find the joint distribution of (X1, X2) GIVEN X3=X3 C. P(max(X1,X2)<X3) D. P(X1>X2|X3=1)arrow_forwardYou currently own a portfolio of eight stocks. Using the Markowitz model, you computed the optimal mean/variance portfolio. Theweights of these two portfolios are shown in the following table: Stock A B C D E F G HYour Portfolio 0.12 0.15 0.13 0.10 0.20 0.10 0.12 0.08M/V Portfolio 0.02 0.05 0.25 0.06 0.18 0.10 0.22 0.12 You would like to rebalance your portfolio in order to be closer to the M/V portfolio. To avoid excessively high transaction costs, you decide to rebalance only three stocks from your portfolio. Let xi denote the weight of stock i in your rebalanced portfolio. The objective is to minimize the quantity |x1-0.02|+|x2-0.05|+|x3-0.25|+...+|x8-0.12| which measures how closely the rebalanced portfolio matches the M/V portfolio. Formulate this problem as a mixed integer linear program.arrow_forwardDevore [2010] Consider a system consisting of three components as pictured. The system will continue to function as long as the first component functions and either component 2 or component 3 functions. Let X1, X2, and X3 denote the lifetimes of components 1, 2, and 3, respectively. Suppose the Xi ’s are independent of one another and each Xi has an exponential distribution with parameter λ. (a) Let Y denote the system lifetime. Obtain the cumulative distribution function of Y and differentiate to obtain the pdf. [Hint: F(y) = P(Y ≤ y); express the event {Y ≤ y} in terms of unions and/or intersections of the three events {X1 ≤ y}, {X2 ≤ y}, and {X3 ≤ y}.] (b) Compute the expected system lifetime.arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning