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# Let U ~ U (0, 1). Let a and b be constants with aa. Find

## Problem 17E Chapter 4.8

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition

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Problem 17E

Let U ~ U (0, 1). Let a and b be constants with a

a. Find the cumulative distribution function of U (use the result of Exercise 16).

b. Show that P(X≤ x) = P(U≤  (x-a)/(b - a)).

c. Use the result of part (b) to show that X ~ U (a, b).

Step-by-Step Solution:

Step 1 of 4:

Given,

Let x be a continuous random variable with probability density function

f(x) =  ,

We have U~U(0,1)

That is, f(x) = 1

Step 2 of 4:

The claim is to find the cumulative distribution function of U

From the definition of cdf

F(x) = f(x) dx

If x<0, we have that f(x) =0, so F(x) = 0

If, 0 < x  1

F(x) = f(x) dx

Where, f(x) =  ,

F(x) =dx

=  (x

= x

Therefore, P(X  x) = x,  0 < x  1

If, x = 1

F(x) = f(x) dx

=  x

=  1

Hence, the cumulative distribution of x is

Step 3 of 4

Step 4 of 4

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Let U ~ U (0, 1). Let a and b be constants with aa. Find

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Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 4.8 - Problem 17e

Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 4.8 - Problem 17e