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Chapter 8: Q135E (page 452)

4.135 Suppose xhas an exponential distribution with θ=1. Find

the following probabilities:

a.P(x>1)b.P(x≤3)cP(x>1.5)d.P(x≤5)

Short Answer

Expert verified
  1. .The probability is 0.3679
  2. The probability is 0.9502
  3. The probability is 0.2231
  4. The probability is 0.9933

Step by step solution

01

Given Information

The random variable x has an exponential distribution withθ=1

02

The probability density function (PDF) of x

Here x is a random variable with parametersθ=1

The pdf of x is given by,

f(x,θ)=1θexp-xθ,x>0

Here, θ=1

f(x)=exp(-x);x>0

03

Finding cdf of x

F(x)=P(X≤x)=∫0xf(t)dt=∫0xexp(-t)dt=exp(-t)-10x=-exp(-x)+1=1-exp(-x)F(x)=1-exp(-x)

04

Finding the probability when P(x > 1)

a.P(x>1)=1-P(x≤1)=1-F(1)=1-1-exp(-1)=exp(-1)=0.37879=0.3679

Thus, the required probability is 0.3679.

05

Finding the probability when P(x≤3)

b.P(x≤3)=F(3)=1-exp(-3)=1-0.049787=0.950213=0.9502

Thus, the required probability is 0.9502.

06

Finding the probability when P(x>1.5)

c.P(x>1.5)=1-P(x≤1.5)=1-F(1.5)=1-1-exp(-1.5)=exp(-1.5)=0.22313=0.2231

The required probability is 0.2231.

07

Finding the probability when P(x≤5)

d.P(x≤5)=F(5)=1-exp(-5)=1-0.006738=0.993262=0.9933

The required probability is0.9933.

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Most popular questions from this chapter

Question: Performance ratings of government agencies. The U.S. Office of Management and Budget (OMB) requires government agencies to produce annual performance and accounting reports (PARS) each year. A research team at George Mason University evaluated the quality of the PARS for 24 government agencies (The Public Manager, Summer 2008), where evaluation scores ranged from 12 (lowest) to 60 (highest). The accompanying file contains evaluation scores for all 24 agencies for two consecutive years. (See Exercise 2.131, p. 132.) Data for a random sample of five of these agencies are shown in the accompanying table. Suppose you want to conduct a paired difference test to determine whether the true mean evaluation score of government agencies in year 2 exceeds the true mean evaluation score in year 1.

Source: J. Ellig and H. Wray, “Measuring Performance Reporting Quality,” The Public Manager, Vol. 37, No. 2, Summer 2008 (p. 66). Copyright © 2008 by Jerry Ellig. Used by permission of Jerry Ellig.

a. Explain why the data should be analyzedusing a paired difference test.

b. Compute the difference between the year 2 score and the year 1 score for each sampled agency.

c. Find the mean and standard deviation of the differences, part

b. Use the summary statistics, part c, to find the test statistic.

e. Give the rejection region for the test using a = .10.

f. Make the appropriate conclusion in the words of the problem.

A paired difference experiment yielded ndpairs of observations. In each case, what is the rejection region for testing H0:μd>2?

a. nd=12,α=.05

b.nd=24,α=.10

c.nd=4,α=.025

d.nd=80,α=.01

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a. What is the probability that between 55% and 65% of the sampled adults work during summer vacation?

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