# Thread: Variance for Functions

1. ## Variance for Functions

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

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

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

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

and variance:

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

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