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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.6: Exponential And Logarithmic Equations

Problem 69E

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Transcribed Image Text:a) A random variable X has an exponential distribution with probability density function given by f(x) = {ledx; for x 2 0 lo ,elsewhere Find the 75th percentile of X. b) A certain brand of light bulb has a lifespan which is exponentially distribution with a mean of 35 days. 1. What is the probability that a randomly selected light bulb of this brand will last more than 45 days? II. If it is known that a light bulb has already lasted more than 25 days, what is the probability that it will at least last for an additional 50 days. (Hint: Memoryless Property) c) Suppose that the amount of time it takes for someone to finish a task, denoted by X, has the uniform distribution between 20 and 80 minutes. 1. Write a suitable probability density function for the time it takes someone to finish the task. II. Using the density function in (1) above derive the mean of X. II. Derive the variance of X using the results obtained in (I) and (I) above. IV. Find the probability that an individual takes between 30 and 75 minutes to finish the task.

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The operator of a pumping station has observed that demand for waterduring early afternoon hours has an approximately exponential distribution with mean 1000cfs (cubic feet per second).a) Find the probability that the demand will exceed 700 cfs during the early afternoonon a randomly selected day.b) What water-pumping capacity should the station maintain during early afternoons sothat the probability that demand will be below the capacity on a randomly selectedday is 0.995?c) Of the three randomly selected afternoons, what is the probability that on at least twoafternoons the demand will exceed 700 cfs? 2. Let Y1 and Y2 be random variables with joint density functionf(y1, y2) = (6/7(y^2+y1y2/2) 0 < y1 < 1, 0 < y2 < 2,0, elsewherea) Find marginal density functions. Are Y1 and Y2 independent?b) Find P(0 < Y1 < 0.3, −2 < Y2 < 1).c) Find P(0.6 < Y1 < 1|0 < Y2 < 1). 3.The joint density function of Y1 and Y2 is given byf(y1, y2) = (y1 + y2), 0 <…
The operator of a pumping station has observed that demand for water during early afternoon hours has an approximately exponential distribution with mean 1000 cfs (cubic feet per second). b) What water-pumping capacity should the station maintain during early afternoons so that the probability that demand will be below the capacity on a randomly selected day is 0.995? c) Of the three randomly selected afternoons, what is the probability that on at least two afternoons the demand will exceed 700 cfs?
Suppose the time it takes Alex to do this exam is exponentially distributed with parameter 3 per hour, and the time it takes Ben to do the exam is exponentially distributed with parameter 2per hour. Assume that these two times are independent.(a) What is the probability that Alex finishes before Ben?(b) What is the expected time in minutes until the first one finishes this exam?(c) What is the probability that neither Alex nor Ben finishes the exam within 3 hours?

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