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Chapter 8: Q132E (page 452)

4.132 Suppose xis a random variable best described by a uniform

probability distribution with c= 3 and d= 7.

a. Find f(x)

b. Find the mean and standard deviation of x.

c. FindP(渭-蟽鈮�x鈮�渭+蟽)

Short Answer

Expert verified

a. The probability density function is

f(x)=0.253鈮�x鈮�70;otherwise

b. The mean is 5 and standard deviation is 1.1547.

c.P(渭-蟽鈮�x鈮�渭+蟽)

Step by step solution

01

Given Information

Here, x is a uniform random variable with parameters c=3 and d=7.

02

Finding the pdf of x

a.

The probability density function random variable x is given by

f(x)=1d-c;c<x<d

Here, c=3 and d=7.

So, the pdf of x is:

f(x)=17-3=14=0.25

Thus, f (x) = 0.25 ; 3 < x < 7

03

Finding the mean and standard deviation of x.

b.

The mean of the random variable x is given by,

渭=c+d2=3+72=102=5

The standard deviation of x is given by,

蟽=d-c12=7-312=423=23=1.1547

Thus, the mean 渭=5and standard deviation 蟽=1.1547.

04

Finding the P(μ-σ≤x≤μ+σ)

c.P(渭-蟽鈮�x鈮�渭+蟽)=鈭�渭-蟽渭+蟽f(x)dx=鈭�渭-蟽渭+蟽0.25dx=0.25鈭�渭-蟽渭+蟽dx=0.25x渭-蟽渭+蟽=0.25渭+蟽-渭+蟽=0.25脳2蟽=0.50脳23=0.5774

So, the required probability is 0.5774.

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Most popular questions from this chapter

Question: Two independent random samples have been selected鈥�100 observations from population 1 and 100 from population 2. Sample means x炉1=26.6,x炉2= 15.5 were obtained. From previous experience with these populations, it is known that the variances are蟽12=9and蟽22=16 .

a. Find 蟽(x炉1-x炉2).

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DM Dosage

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a.Ha:蟽2鈮�25;伪=0.5;n=16

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