Let Z N(0, 1), and let X = Z + where and > 0 are constants. Let represent the cumulative

Chapter 4, Problem 30

(choose chapter or problem)

Let \(Z \sim N(0,1)\), and let \(X=\sigma Z+\mu\) where \(\mu \text { and } \sigma>0\) are constants. Let \(\Phi\) represent the cumulative distribution function of , and let \(\varphi\) represent the probability density function, so \(\varphi(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\)

a. Show that the cumulative distribution function of  is \(F_{X}(x)=\Phi\left(\frac{x-\mu}{\sigma}\right)\)

b. Differentiate \(F_{X}(x)\) to show that \(X \sim N\left(\mu, \sigma^{2}\right)\)

c. Now let \(X=-\sigma Z+\mu\). Compute the cumulative distribution function of  in terms of \(\Phi\), then differentiate it to show that \(X \sim N\left(\mu, \sigma^{2}\right)\).

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Text Transcription:

Z \sim N(0,1)

X=\sigma Z+\mu

\mu  and  \sigma>0

\Phi

\varphi

\varphi(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}

F_{X}(x)=\Phi\(\frac{x-\mu}{\sigma})

F_{X}(x)

X \sim N (\mu, \sigma^{2}\)

X=-\sigma Z+\mu

\Phi

X \sim N (\mu, \sigma^{2})