Self-Study: Function of a Gaussian RV

I am a beginner, solving a preparatory examen to study, and I have the following problem, where i don't understand how to start to find the answer.

Is it a transformation one to one, or not? I'm trying to find out if I should derivate, or maybe there is a method i should use.

It would be nice to get some advice on how to attack the problem.

**I would appreciate any book or link over the topic **

Thank you

• You are asked to find a transform of X with mean zero and variance $1$. Maybe start with finding a transform with mean zero... (Any book covering the Normal distribution will do. And so will the Wikipedia entry on the Normal distribution.) – Xi'an Jan 7 at 15:49

You are heading in the wrong direction with your line of thinking.

Just off the top of your head, if you had a random variables with a known mean and variance, what transformation would you use to center/scale that random variable so that it has mean 0 and variance 1?

You don't need a book or link, just two mathfacts:

$$E[aX+b] = aE[X] + b$$

$$\text{var}[aX+b] = a^2\text{var}[X]$$

• Ok @AdamO, if i have: g(${\sqrt{2\pi} \sigma}e^{-\frac{x^2}{2 \sigma^2}}$) = ${\sqrt{2\pi}}e^{-\frac{x^2}{2}}$ , how do i find my g, i'm lost there – mjginno Jan 7 at 16:45
• @mjginno A good text is "Probability Models" by Sheldon Ross. The error you have here is confusing the density with the actual random variable. – AdamO Jan 7 at 16:53
• @mjginno if a random variable $X$ has a density $f$, $X+1$ has a density but it's not $f+1$. Rather, you have to use a change of variable method, with either the Jacobian, CDF-method, or moment generating function $E(e^{tx})$ to understand what the actual density will be. This is a very hard first step into probability, but when the base facts are proven, it gets easier. Namely if $X, Y$ are normal then $aX + bY$ is also normal. – AdamO Jan 7 at 16:58
• Thanks Adam, that's very useful, at least i know I have to gain the skills to use any of the methods correctly. Great help. – mjginno Jan 7 at 17:02