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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,

μ=c+d2=100+2002=150

The standard deviation is given by,

σ=d-c12=200-10012=10012=28.8675

Thus, the mean μ=150and standard deviation σ=28.8675 .

03

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

a.

Px>μ+2σ=Px>150+2×28.8675=Px>207.735=∫207.735∞fxdx=∫207.735∞0dx=0

Thus, the required probability is 0.

04

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

Pμ-2σ<x<μ+2σ=P150-2×28.8675<x<150+2×28.8675=P92.265<x<207.735=∫92.265207.735fxdx=∫1002000.001dx=0.01x100200=0.01×200-100=0.01×100=1

Thus, the required probability is 1.

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