Math

StatisticsProbability and Statistics for Engineering and the Sciencesa. Show that if X has a normal distribution with parameters µ and σ , then Y = aX + b (a linear function of X ) also has a normal distribution. What are the parameters of the distribution of Y [i.e., E ( Y ) and V ( Y )]? [ Hint : Write the cdf of Y , P ( Y ≤ y ), as an integral involving the pdf of X , and then differentiate with respect to y to get the pdf of Y .] b. If, when measured in °C, temperature is normally distributed with mean 115 and standard deviation 2, what can be said about the distribution of temperature measured in °F?BuyFind*arrow_forward*

9th Edition

Jay L. Devore

Publisher: Cengage Learning

ISBN: 9781305251809

Chapter 4.3, Problem 57E

Textbook Problem

**a.** Show that if *X* has a normal distribution with parameters *µ* and *σ*, then *Y* = *aX* + *b* (a linear function of *X*) also has a normal distribution. What are the parameters of the distribution of *Y* [i.e., *E*(*Y*) and *V*(*Y*)]? [*Hint*: Write the cdf of *Y*, *P*(*Y* ≤ *y*), as an integral involving the pdf of *X*, and then differentiate with respect to *y* to get the pdf of *Y*.]

**b.** If, when measured in °C, temperature is normally distributed with mean 115 and standard deviation 2, what can be said about the distribution of temperature measured in °F?

Expert Solution

**a.**

To determine

Prove that *Y* has a normal distribution.

Find the parameters of *Y*.

**Given info:**

The random variable *X* has a normal distribution with parameters *X* is *Y*, such that

**Calculation:**

The cumulative distribution function (cdf) of the random variable *Y* at *y* is:

The probability density function (pdf) of *Y*, *Y* with respect to *y*.

Now, pdf of the normal random variable *X* with parameters

Replace *X*:

Expert Solution

**b.**

To determine

Describe the distribution of the temperature measured in

Probability and Statistics for Engineering and the Sciences

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