Math Help - Variance for Functions

1. Variance for Functions

I have never seen anyone do this but has anyone defined the variance for a countinous function $f(x)$ on interval $[a,b]$?
I used "reasonable" definition to get:
$\frac{1}{b-a}\int^b_a[f(x)-\= x]^2dx$
where,
$\=x$ is the average value, thus,
$\=x=\frac{1}{b-a}\int^b_a f(x)dx$

2. Originally Posted by ThePerfectHacker
I have never seen anyone do this but has anyone defined the variance for a countinous function $f(x)$ on interval $[a,b]$?
I used "reasonable" definition to get:
$\frac{1}{b-a}\int^b_a[f(x)-\= x]^2dx$
where,
$\=x$ is the average value, thus,
$\=x=\frac{1}{b-a}\int^b_a f(x)dx$
Yes, usually in the following context:

Let $X$ be a random variable with pdf $p(x)$, and $f$ be a function defined
on the event space $\Omega$ of $X$. Then $Y=f(X)$, is a random variable with mean:

$\mu _Y = \int_{\Omega} f(x)p(x) dx$,

and variance:

$\sigma _Y^2 =\int_{\Omega}(f(x)- \mu_Y)^2 p(x) dx$

RonL

3. CaptainBlack, I can see you are very knowledgable in Statistics.

4. Originally Posted by ThePerfectHacker
CaptainBlack, I can see you are very knowledgable in Statistics.

Mebee

RonL