2
$\begingroup$

I am trying to find the sum of n uniformly distributed i.i.d random variables in the range [0-W]. I am aware that if the variables are distributed in the interval (0,1) then their convolution is given by Irwin-Hall distribution. But what if the random variables are uniformly distributed over the interval [0-W]?

$\endgroup$
3
$\begingroup$

If we replace uniforms on $(0,1)$ by uniforms on $(0,w)$, the resulting random variable $Y$ has the same distribution as $wX$, where $X$ has Irwin-Hall distribution. In particular, $\Pr(Y\le y)=\Pr(wX\le y)=\Pr(X\le \frac{y}{w})$. It follows that if $f_X$ is the density function of the Irwin-Hall, then $Y$ has density $f_Y(y)=\frac{1}{w}f_X(y/w)$.

In a similar way, one can read off almost any desired facts about the distribution of $Y$ from related facts about the Irwin-Hall.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.