3

If you have a random variable which takes on values in the range $[a, b]$, is it necessary that the variance be in the range $[a, b]$?

What is the range of variance in general?

I feel like this is important to understanding it intuitively as a measure of ``how spread out the data is".

jaynp
  • 2,151

3 Answers3

5

For $[a,b]=[0,1]$, this question already has some great answers here:

Now for the general case, start with $X$ supported on $[0,1]$ and shift it to $Y=a+(b-a)X$ supported on $[a,b]$; then $\mathrm{Var}(Y)=\mathrm{Var}(X+(b-a)X)=\mathrm{Var}((b-a)X)=(b-a)^2\mathrm{Var}(X)$. So we have $0\leq \mathrm{Var}(Y)\leq \frac14(b-a)^2$.

Community
  • 1
Chris Culter
  • 26,806
  • I see how you prove $\mathrm{Var}(Y) = (b-a)^2 \mathrm{Var}(X)$. How does the inequality below follow from that? – jaynp Aug 14 '13 at 17:26
  • 1
    @user1850672 Did you read the links I provided? We already have $0\leq\mathrm{Var}(X)\leq\frac14$. – Chris Culter Aug 14 '13 at 17:35
2

Suppose the range of $X$ is the interval $[a,b]$. For any constant $c$, let $Y=X+c$. The variance of $Y$ is the same as the variance of $X$, but the range of $Y$ is now $[a+c,b+c]$.

We can therefore shift our random variable to the interval $[0,b-a$ without changing the variance. Now since the mean $\mu$ is in the interval, and the variance is $E(X-\mu)^2$, it is clear that the variance is bounded.

André Nicolas
  • 507,029
  • The question does not ask whether "the variance is bounded" (whatever that means) but what is its range. (@Upvoters: Can you explain?) – Did Aug 14 '13 at 21:41
  • I did not address the range of variance, since by the time I was starting to write that part, someone else had posted. The first thing I wrote was an answer to the first question, which asked whether the range was in the interval $[a,b]$. – André Nicolas Aug 14 '13 at 21:47
1

No, but close. If a random variable takes on values in the range $[a,b]$ the maximum variance is achieved when it is equal to $a$ half the time and $b$ the other half, so we can the variance is in the range $[0,(\frac{b-a}{2})^2]$. On the other hand, a random variable not constrained to an interval may not have a finite variance.

Dan Brumleve
  • 17,796