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Chapter 4: Q136E (page 276)

4.135 Suppose xhas an exponential distribution with胃=2.5 . Find

the following probabilities:

a.P(x鈮�4)b.P(x>5)c.P(x鈮�2)d.P(x>3)

Short Answer

Expert verified

a. The probability is 0.7981

b. The probability is 0.1353

c. The probability is 0.5507

d. The probability is 0.3012

Step by step solution

01

Given Information

The random variable x has an exponential distribution with胃=2.5

02

The probability density function of x

Here x is a random variable with parameters胃=2.5

The pdf of x is given by,

fx;胃=1胃exp-x胃;x>0

Here,

fx=12.5exp-x2.5;x>0

03

Finding cdf of x

Fx=PX鈮�x=鈭�0xftdt=鈭�0x12.5exp-t2.5dt=12.5exp-t2.512.50x=exp-x2.5+1=1-exp-x2.5=1-exp-0.4xFx=1-exp-0.4x

04

Finding the probability when P(x≤4)

a.

Px>4=F4=1-exp-0.4脳4=1-exp--1.6=1-0.201897=0.798103~0.7981

Thus, the required probability is 0.7981.

05

Finding the probability when (x>5)

b.

Px>5=1-Px鈮�5=1-F5=1-1-exp-0.4脳5=exp-2=0.135335~0.1353

Thus, the required probability is 0.1353.

06

Finding the probability when P(x≤2)

c.

Px鈮�2=F2=1-exp-0.4脳2=1-exp-0.8=1-0.449329=0.550671~0.5507

Thus, the required probability is 0.5507.

07

Finding the probability when (x>3)

d.

Px>3=1-Px鈮�3=1-F3=1-1-exp-0.4脳3=exp-1.2=0.301194~0.3012

Thus, the required probability is 0.3012.

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Most popular questions from this chapter

NHTSA crash tests. Refer to the NHTSA crash tests of new car models, Exercise 4.3 (p. 217). A summary of the driver-side star ratings for the 98 cars in the file is reproduced in the accompanying Minitab printout. Assume that one of the 98 cars is selected randomly and let x equal the number of stars in the car鈥檚 driver-side star rating.

  1. Use the information in the printout to find the probability distribution for x.
  2. FindP(x=5).
  3. FindP(x鈮�2).
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Waiting for a car wash. An automatic car wash takes exactly 5 minutes to wash a car. On average, 10 cars per hour arrive at the car wash. Suppose that 30 minutes before closing time, 5 cars are in line. If the car wash is in continuous use until closing time, will anyone likely be in line at closing time?

Suppose the random variable x is best described by a normal distribution with 渭=30 and 蟽=4. Find the z-score that corresponds to each of the following x values:

a.x=20b.x=30c.x=2.75d.x=15e.x=35f.x=25

Testing for spoiled wine. Suppose that you are purchasing cases of wine (12 bottles per case) and that, periodically, you select a test case to determine the adequacy of the bottles鈥� seals. To do this, you randomly select and test 3 bottles in the case. If a case contains 1 spoiled bottle of wine, what is the probability that this bottle will turn up in your sample?

Privacy and information sharing. Refer to the Pew Internet & American Life Project Survey (January 2016), Exercise 4.48 (p. 239). The survey revealed that half of all U.S. adults would agree to participate in a free cost-saving loyalty card program at a grocery store, even if the store could potentially sell these data on the customer鈥檚 shopping habits to third parties. In a random sample of 250 U.S. adults, let x be the number who would participate in the free loyalty card program.

a. Find the mean of x. (This value should agree with your answer to Exercise 4.48c.)

b. Find the standard deviation of x.

c. Find the z-score for the value x = 200.

d. Find the approximate probability that the number of the 250 adults who would participate in the free loyalty card program is less than or equal to 200.
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