Let $X$ be a random variable having expected value $\mu$ and variance $\sigma^2$. Find the Expected Value and Variance of $Y = \frac{X−\mu}{\sigma}$.
I would like to show some progress I've made so far, but honestly I've been thinking about this problem for the past few days but just have no idea where to start. Any hint or insight on a starting point would be much appreciated.
Thanks!
Best Answer
Given a random variable $X$, a location scale transformation of $X$ is a new random variable $Y=aX+b$ where $a$ and $b$ are constants with $a>0$.
The location scale transformation $aX+b$ horizontally scales the distribution of $X$ by the factor $a$, and then shifts the distribution so obtained by the factor $b$ on the real line $\mathbb{R}$.
Now, here is an hint to your problem: $Y=\dfrac{X-\mu}{\sigma}=\dfrac{1}{\sigma}X-\dfrac{\mu}{\sigma}$, which can be written as $aX+b$. Find $a$ and $b$, and then use the location-scale transformation.