1. ## "Continuous" binomial distribution

Hi.

I want to use some kind of "continuous" binomial distribution. I know binomial distribution is defined for natural numbers, but I want to extend it to the use of real numbers.

I also know that the gamma function is an extension of factorial to be used with real numbers.

My question is:

Is $f(x) = \frac{\Gamma(n+1)}{\Gamma(x+1)*\Gamma(n+1-x)} p^x (1-p)^{n-x}$ a probability density function? More precisely, is the integral from 0 to n equal to 1?

Playing with excel it doesn't seem to be 1, but i'm not sure. Do you know what function should i use?

Thanks.

2. You may want to look at a Beta random variable.
http://en.wikipedia.org/wiki/Beta_distribution
If you want to extend it's support from (0,1) to say (0,n) you can tranform it by letting W=nX.

3. Thanks! That's what I was looking for!

4. The Beta is a generalization of a U(0,1) rv.
If you let $\alpha=\beta=1$ you get a U(0,1).
Do you know how to transform a rv, so the support is (0,n) instead?