9th Edition

ISBN: 9780321794772

Author: Sheldon Ross

Publisher: PEARSON

Concept explainers

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could co…

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a…

Videos

Textbook Question

Chapter 5, Problem 5.23TE

Compute the hazard rate function of a Weibull random variable and show it is increasing when β ≥ 1 and decreasing when β ≤ 1 .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5, Problem 5.22TE

Chapter 5, Problem 5.24TE

Students have asked these similar questions

Suppose the variance of the difference of two random variables X and Y is zero. Then, show that X = Y + c for some constant c almost surely. Also find the value of c.

Suppose that Y is a discrete random variable with mean u and variance ot and let W = 5Y. (a) Do you expect the mean of W to be larger than, smaller than, or equal to µ = E(Y)? Why? O The mean of W will be larger than the mean of Y if u 0 and equal to u if u = 0. O The mean of W will be larger than the mean of Y if u > 0, smaller if u 0. O The mean of W will be larger than the mean of Y if u = 0, smaller if u > 0 and equal to u if u 0. (b) Use the theorem below to express E(W) = E(5Y) in terms of u, where u = E(Y). Let Y be a discrete random variable with probability function p(v), g(Y) be a function of Y, and c be a constant. Then E[cg(Y)] = cE[g(Y)]. E(W) = Does this result agree with your answer to part (a)? O Yes O No (c) Recalling that the variance is a measure of spread or dispersion, do you expect the variance of W to be larger than, smaller than, or equal to o? = V(Y)? Why? O The variance of W will be smaller than o, since the spread of values of W has decreased. O The…

Let X1 be a normal random variable with mean 2 and variance 3 and let X2 be a normal random variable with mean 1 and variance 4. Assume that X1 and X2 are independent. (a) What is the distribution of the linear combination Y = 2X1 +3X,? (b) What is the distribution of the linear combination Y = X, – X,?

Chapter 5 Solutions

A First Course in Probability

Ch. 5 - Let X be a random variable with probability...Ch. 5 - Prob. 5.2P Ch. 5 - Prob. 5.3P Ch. 5 - The probability density function of X. the...Ch. 5 - Prob. 5.5P Ch. 5 - Compute E[X] if X has a density function given by...Ch. 5 - The density function of X is given by...Ch. 5 - The lifetime in hours of an electronic tube is a...Ch. 5 - Consider Example 4b &I of Chapter 4 &I, but now...Ch. 5 - Trains headed for destination A arrive at the...

Ch. 5 - A point is chosen at random on a line segment of...Ch. 5 - A bus travels between the two cities A and B....Ch. 5 - You arrive at a bus stop at 10A.M., knowing that...Ch. 5 - Let X be a uniform (0, 1) random variable. Compute...Ch. 5 - If X is a normal random variable with parameters...Ch. 5 - The annual rainfall (in inches) in a certain...Ch. 5 - The salaries of physicians in a certain speciality...Ch. 5 - Suppose that X is a normal random variable with...Ch. 5 - Let be a normal random variable with mean 12 and...Ch. 5 - If 65 percent of the population of a large...Ch. 5 - Suppose that the height, in inches, of a...Ch. 5 - Every day Jo practices her tennis serve by...Ch. 5 - One thousand independent rolls of a fair die will...Ch. 5 - The lifetimes of interactive computer chips...Ch. 5 - Each item produced by a certain manufacturer is,...Ch. 5 - Two types of coins are produced at a factory: a...Ch. 5 - In 10,000 independent tosses of a coin, the coin...Ch. 5 - Twelve percent of the population is left handed....Ch. 5 - A model for the movement of a stock supposes that...Ch. 5 - An image is partitioned into two regions, one...Ch. 5 - a. A fire station is to be located along a road of...Ch. 5 - The time (in hours) required to repair a machine...Ch. 5 - If U is uniformly distributed on (0,1), find the...Ch. 5 - Jones figures that the total number of thousands...Ch. 5 - The lung cancer hazard rate (t) of a t-year-old...Ch. 5 - Suppose that the life distribution of an item has...Ch. 5 - If X is uniformly distributed over (1,1), find (a)...Ch. 5 - Prob. 5.38P Ch. 5 - If X is an exponential random variable with...Ch. 5 - Prob. 5.40P Ch. 5 - Find the distribution of R=Asin, where A is a...Ch. 5 - Let Y be a log normal random variable (see Example...Ch. 5 - The speed of a molecule in a uniform gas at...Ch. 5 - Show that E[Y]=0P{Yy}dy0P{Yy}dy Hint: Show that...Ch. 5 - Show that if X has density function f. then...Ch. 5 - Prob. 5.4TE Ch. 5 - Use the result that for a nonnegative random...Ch. 5 - Prob. 5.6TE Ch. 5 - The standard deviation of X. denoted SD(X), is...Ch. 5 - Let X be a random variable that takes on values...Ch. 5 - Show that Z is a standard normal random variable;...Ch. 5 - Let f(x) denote the probability density function...Ch. 5 - Let Z be a standard normal random variable Z and...Ch. 5 - Use the identity of Theoretical Exercises 5.5 .Ch. 5 - The median of a continuous random variable having...Ch. 5 - The mode of a continuous random variable having...Ch. 5 - If X is an exponential random variable with...Ch. 5 - Compute the hazard rate function of X when X is...Ch. 5 - If X has hazard rate function X(t), compute the...Ch. 5 - Prob. 5.18TE Ch. 5 - If X is an exponential random variable with mean...Ch. 5 - Prob. 5.20TE Ch. 5 - Prob. 5.21TE Ch. 5 - Compute the hazard rate function of a gamma random...Ch. 5 - Compute the hazard rate function of a Weibull...Ch. 5 - Prob. 5.24TE Ch. 5 - Let Y=(Xv) Show that if X is a Weibull random...Ch. 5 - Let F be a continuous distribution function. If U...Ch. 5 - If X is uniformly distributed over (a,b), what...Ch. 5 - Consider the beta distribution with parameters...Ch. 5 - Prob. 5.29TE Ch. 5 - Prob. 5.30TE Ch. 5 - Prob. 5.31TE Ch. 5 - Let X and Y be independent random variables that...Ch. 5 - Prob. 5.33TE Ch. 5 - The number of minutes of playing time of a certain...Ch. 5 - For some constant c. the random variable X has the...Ch. 5 - Prob. 5.3STPE Ch. 5 - Prob. 5.4STPE Ch. 5 - The random variable X is said to be a discrete...Ch. 5 - Prob. 5.6STPE Ch. 5 - To be a winner in a certain game, you must be...Ch. 5 - A randomly chosen IQ test taker obtains a score...Ch. 5 - Suppose that the travel time from your home to...Ch. 5 - The life of a certain type of automobile tire is...Ch. 5 - The annual rainfall in Cleveland, Ohio, is...Ch. 5 - Prob. 5.12STPE Ch. 5 - Prob. 5.13STPE Ch. 5 - Prob. 5.14STPE Ch. 5 - The number of years that a washing machine...Ch. 5 - Prob. 5.16STPE Ch. 5 - Prob. 5.17STPE Ch. 5 - There are two types of batteries in a bin. When in...Ch. 5 - Prob. 5.19STPE Ch. 5 - For any real number y define byy+=y,ify00,ify0 Let...Ch. 5 - With (x) being the probability that a normal...Ch. 5 - Prob. 5.22STPE

Knowledge Booster

$Probability$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.

Compute the hazard rate function of a Weibull random variable and show it is increasing when β ≥ 1 and decreasing when β ≤ 1 .

Solution Summary: The author explains the hazard rate function of a Weibull random variable.

Concept explainers

Videos

Want to see the full answer?

Chapter 5 Solutions