- #1

bonfire09

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## Homework Statement

Let ##X## have the pdf ##f_X(x)=\dfrac{1}{\sqrt{2\pi\sigma^2}}e^{\dfrac{-(x-\mu)^2}{2\sigma^2}}## where ##-\infty<x<\infty,-\infty<\mu<\infty,\sigma>0##. Let ##Z=g(X)=\frac{X-\mu}{\sigma}##. Find the pdf and cdf of ##Z##[/B]

## Homework Equations

## The Attempt at a Solution

Basically I noticed that ##X## has a normal distribution. I get for the cdf of ##Z## is ##P((\dfrac{x-\mu}{\sigma})\leq t) ## ##=P(x-\mu\leq t\sigma)=P(x\leq\mu+t\sigma)=F_Z(\mu+t\sigma)=\int_{-\infty}^{\mu+t\sigma} \dfrac{1}{\sqrt{2\pi\sigma^2}}e^{\dfrac{-(x-\mu)^2}{2\sigma^2}} dx##. The pdf of ##Z## I get that ##f_z(x)=F'_Z(t)=\dfrac{1}{\sqrt{2\pi\sigma^2}}e^{\dfrac{-(x-\mu)^2}{2\sigma^2}} *\sigma##. Not sure if I am on the right track. Thanks.