Answer: a. Show that if X has a normal distribution with

Chapter 4, Problem 57E

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QUESTION:

Problem 57E

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?

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QUESTION:

Problem 57E

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?

ANSWER:

Answer:

Step1 of 2:

a). To show that if X has a normal distribution with parameters $\mu{}$ and $\sigma{}$ .

Then the linear function of x is

Y = aX + b this is also a normal distribution.

Here we need to find the parameters of the distribution Y.

Then, x follows a normal distribution with mean $\mu{}$ and variance $\sigma{}$ .

That is,

$X\sim{}N(\mu{},\sigma{})$

The pdf of ‘x’ is

${f}_{x}^{\ }(x)$ = $\frac{1}{\sqrt{2\Pi{}}\ \sigma{}}e{{\ }^{-\frac{({x-\mu{})}^{2}}{{2\sigma{}}^{2}}}}^{\ }$ ; - $\infty{}<x<\infty{}$

Differentiate on both sides

We get,

${f}_{x}^{\ }(x)$ dx = $\frac{1}{\sqrt{2\Pi{}}\ \sigma{}}e{{\ }^{-\frac{({x-\mu{})}^{2}}{{2\sigma{}}^{2}}}}^{\ }$ dx ……..(1)

Given the linear function is

Y = aX + b

$\Rightarrow{}$ Y - b = aX

$\Rightarrow{}$ x = $\frac{y-b}{a}$

Let x = $\frac{y-b}{a}$ = g(y)

Substitute the ‘x’ value in equation (1).

${f}_{x}^{\ }(g(y))$ dg(y) = $\frac{1}{\sqrt{2\Pi{}}\ \sigma{}}e{{\ }^{-\frac{(\frac{y-b}{a}{-\mu{})}^{2}}{{2\sigma{}}^{2}}}}^{\ }$ dg(y)

${f}_{x}^{\ }(g(y))$ dg(y) = $\frac{1}{\sqrt{2\Pi{}}\sigma{}}e{\ }^{-(\frac{(y-b-a\mu{}{)}^{2}}{{2{\sigma{}}^{2}a}^{2}})}.\frac{dy}{a}$

${f}_{y}^{\ }(y)\ d(y)$ = $\frac{1}{\sqrt{2\Pi{}}\sigma{}a}e{\ }^{-(\frac{(y-{\mu{}}_{y}){\ }^{2}}{2{\sigma{}{{\ }_{y}^{2}}_{\ }}^{\ }})\ }dy$ (where, ${\mu{}}_{y}=a{\mu{}}_{x}+b$