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# Let Z ~ N(0, 1), and let X = ?Z +µ where p and ? > 0 are

ISBN: 9780073401331 38

## Solution for problem 30SE Chapter 4

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition

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Problem 30SE

Let Z ~ N(0, 1), and let X = σZ +µ where p and σ > 0 are constants. Let Φ represent the cumulative distribution function of Z, and let Φ represent the probability density function, so

a. Show that the cumulative distribution function of

b. Differentiate Fx(x) to show that X ~ N(µ, σ2).

c. Now let X = -σZ +µ. Compute the cumulative distribution function of X in terms of Φ, then differentiate it to show that X ~ N(µ, σ2).

Step-by-Step Solution:

Step 1 of 3:

(a)

In this question, we are asked to prove the cumulative distribution function of

Let , and let  and is constants.

represent cumulative distribution function of , let  represent the probability density function, so

We can write cumulative distribution function of  random variable  as

……….(1)

Given

Put the value of into the equation (1), this method is also called change a variable method

………..(2)

We can write cumulative distribution function of  random variable  as

……….(3)

equate equation (2) and (3), we can deduce the value of

Put the value of z into the equation (3), we will get,

………….(4)

From equation (2) and (4),we get

=

Hence proved.

Step 2 of 2

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