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.


  • $\begingroup$ What would happen if $X$ were a standard normal? (Ie, $X\sim\mathcal N(0,1)$.) What would happen then? $\endgroup$ Oct 10, 2013 at 4:29

2 Answers 2


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 $\mathbb{R}$.

  • In an intuitive sense, the expected value $\mathbb{E}[X]$ of a random variable is the center of mass of the distribution of $X$. Shifting the distribution of $X$ by a factor $b$, shifts the center of mass by the factor $b$. Scaling the distribution of $X$ by a factor $a$, scales the center of mass by $a$. In other words, $$\mathbb{E}[aX+b]=a\mathbb{E}[X]+b$$
  • Similarly, the variance of $X$ is a measure of the horizontal spread of the distribution of $X$, but the $\text{Var}[X]$ is defined as squared-distance. Thus scaling the distribution of $X$ by a factor $a$, scales the $\text{Var}[X]$ by the factor $a^2$. Shifting the distribution of $X$ by any factor will not affect the spread of distribution, $\text{Var}[X]$, but only affects center of mass. In other words, $$\text{Var}[aX+b]=\text{Var}[aX]=a^2\text{Var}[X]$$

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.


Have you seen the following basic properties of expectation and variance?

(I'd be very surprised if some version of these hadn't been discussed)

$\text{E}(aX+b) = a\text{E}(X)+b$

$\text{Var}(aX+b) = a^2\text{Var}(X)$



If you apply these properties, or better, the versions you'll already have been given, the problem is trivial.

If you still can't see it, try finding $\text{E}(X-\mu)$ first and work from there.


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