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Chapter 4: Q137E (page 276)

4.137 The random variable xis best described by a uniform probability distribution with c= 100 and d= 200. Find the probability that xassumes a value

a. More than 2 standard deviations from

b. Less than 3 standard deviations from

c. Within 2 standard deviations of

Short Answer

Expert verified

a. The probability is 0.

b. The probability is 0.

c. The probability is 1.

Step by step solution

01

Given Information

Here, x is a uniform random variable with parameters c=100 and d=200.

02

Finding the pdf of x

The probability density function random variable x is given by

fx=1d-c;c<x<d

Here, c=100 and d=200

So, the pdf of x is:

fx=1200-100=1100=0.01

Thus,fx=0.01;100<x<2000;otherwise

The mean is given by,

渭=c+d2=100+2002=150

The standard deviation is given by,

蟽=d-c12=200-10012=10012=28.8675

Thus, the mean 渭=150and standard deviation 蟽=28.8675 .

03

Finding the probability when x assumes a value more than 2 standard deviations from μ

a.

Px>渭+2蟽=Px>150+2脳28.8675=Px>207.735=鈭�207.735鈭�fxdx=鈭�207.735鈭�0dx=0

Thus, the required probability is 0.

04

Finding the probability when x assumes a value within 2 standard deviations from μ

P渭-2蟽<x<渭+2蟽=P150-2脳28.8675<x<150+2脳28.8675=P92.265<x<207.735=鈭�92.265207.735fxdx=鈭�1002000.001dx=0.01x100200=0.01脳200-100=0.01脳100=1

Thus, the required probability is 1.

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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.
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0.01 1.21 1.71 2.30 2.96 0.19 1.22 1.75 2.30 2.98 0.51

1.24 1.77 2.41 3.19 0.57 1.48 1.79 2.44 3.25 0.70 1.54

1.88 2.57 3.31 0.73 1.59 1.90 2.61 1.19 0.75 1.61 1.93

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Stock market. Give an example of a continuous random variable that would be of interest to a stockbroker.

The random variable x has a normal distribution with 渭=1000 and 蟽=10.

a. Find the probability that x assumes a value more than 2 standard deviations from its mean. More than 3 standard deviations from .渭

b. Find the probability that x assumes a value within 1 standard deviation of its mean. Within 2 standard deviations of 渭.

c. Find the value of x that represents the 80th percentile of this distribution. The 10th percentile.
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