9
Jan 16

## Scaling a distribution

Scaling a distribution is as important as centering or demeaning considered here. The question we want to find an answer for is this: What can you do to a random variable $X$ to obtain another random variable, say, $Y$, whose variance is one? Like in case of centering, geometric considerations can be used but I want to follow the algebraic approach, which is more powerful.

Hint: in case of centering, we subtract the mean, $Y=X-EX$. For the problem at hand the suggestion is to use scaling: $Y=aX$, where $a$ is a number to be determined.

Using the fact that variance is homogeneous of degree 2, we have

$Var(Y)=Var(aX)=a^2Var(X)$.

We want $Var(Y)$ to be 1, so solving for $a$ gives $a=1/\sqrt{Var(X)}=1/\sigma(X)$. Thus, division by the standard deviation answers our question: the variable $Y=X/\sigma(X)$ has variance and standard deviation equal to 1.

Note. Always use the notation for standard deviation $\sigma$ with its argument $X$.