1The time to failure, t, in hours, of a machine is often exponentially distributed with a probability density function f(t)=ke-kt, 0≤t<∞o, where k = — and a is1athe average amount of time that will pass before a failure occurs. Suppose that the average amount of time that will pass before a failure occurs is 80 hr. Whatis the probability that a failure will occur in 48 hr or less?The probability is(Round to four decimal places as needed.)

To find: The probability that a failure will occur in 48 or less ?

Answered: 1 The time to failure, t, in hours, of…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Suppose that the interval between eruptions of a particular geyser can be modelled by an exponential distribution with an unknown parameter 0 > 0. The probability density function of this distribution is given by f(x; 0) = 0e 0¹, x > 0. The four most recent intervals between eruptions (in minutes) are x₁ = 32, x₂ = 10, x3 = 28, x4 = 60; their values are to be treated as a random sample from the exponential distribution. (a) Show that the likelihood of based on these data is given by L(0) 04-1306 = (b) Show that L'(0) is of the form L'(0) = 0³ e 1300 (4- 1300). (c) Show that the maximum likelihood estimate of 0 based on the data is ~ 0.0308 making your argument clear. (d) Explain in detail how the maximum likelihood estimate of that you have just obtained in part (c) relates to the maximum likelihood estimator of for an exponential distribution.
The life X (in hours) of a battery in constant use is an arbitrary variable with exponential density. What is the probability that the battery will last more than 28 hours if the average life is 8 hours? (Give your answer to four decimal places.) P≈
A company uses wood chips in the manufacture of processed wood pieces of various sizes. Suppose the tons of wood chips T used per day has a probability density function f(T) = 0.25e-0.25T (a) Find the probability of using more than 4 tons of wood chips in a day. (Round your answer to three decimal places.) (b) Find the number of tons 7 (to the nearest ton) so that the probability of using more than 7 tons in a day is 0.07. tons
Do you dislike waiting in line? A supermarket chain has used computer simulation and information technology to reduce the average waiting time for customers at 2,300 stores. Using a new system, which allows the supermarket to better predict when shoppers will be checking out, the company was able to decrease average customer waiting time to just 28 seconds. (a)Assume that supermarket waiting times are exponentially distributed. Show the probability density function of waiting time at the supermarket. f(x) = x ≥ 0 elsewhere (b) What is the probability that a customer will have to wait between 30 and 45 seconds? (Round your answer to four decimal places.) (c) What is the probability that a customer will have to wait more than 2 minutes? (Round your answer to four decimal places.)
(6) The life in years of a certain type of electronic switch, X, has the following probability density function: f(x) = -=-e-2/², x > 0. (a) Find the value of the constant c. And compute the expected value of x. (b) Find the probability that a randomly selected switch will last more than 3 years. In other words, find P(X > 3). (c) Given that a switch has been working for 1 year already, find the probability that the switch will last more than 3 additional years. (d) If 5 of the new switches are installed in different independent systems, what is the probability that there will be 3 of these 5 switches fail during the first three years after the installation?

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