# Defining a distribution

1. Apr 22, 2007

### matfor

Say if I had a random variable $$X$$ that followed a Beta distribution $$B(a,b)$$, and $$a$$ and $$b$$ were random variables.

How would I define the distribution of $$X$$??

2. Apr 22, 2007

### Hurkyl

Staff Emeritus
Well, I imagine you would first want to talk about the joint distribution of X, a, and b. Then you can worry about trying to compute the marginal distribution of X.

3. Apr 22, 2007

### ZioX

Basically Hurkyl said it depends on how a and b are distributed. Once you know this you can calculate the distribution of X using the standard procedures.

4. Apr 22, 2007

### matfor

Why would X be a marginal distribution??
Wikipedia says
But in my situation I do not want to ignore the information from the other variables, ie. variables a and b.

It seems to me that I should look for the conditional distribution ie. $$X|a,b$$??

5. Apr 29, 2007

### Jason Swanson

Let $$f(x,y)$$ be the joint density of $$a$$ and $$b$$. (You can modify the following if one or both of them are discrete.) Then

$$F_X(c) = P(X \le c) = E[P(X \le c | (a,b))].$$

This gives

$$F_X(c) = \int_0^\infty\int_0^\infty P(X \le c | (a,b) = (x,y))f(x,y)\,dx\,dy.$$

By hypothesis, the distribution of $$X$$ given $$(a,b)$$ is Beta. So

$$F_X(c) = \int_0^\infty\int_0^\infty I_c(x,y)f(x,y)\,dx\,dy,$$

where $$c\mapsto I_c(x,y)$$ is the CDF of a Beta random variable with parameters $$x$$ and $$y$$.