How Do You Model a Random Variable with Random Parameters in Its Distribution?

[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 $$X$$??

 

[Hurkyl]

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.

 

[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.

 

[matfor]

Then you can worry about trying to compute the marginal distribution of X

Why would X be a marginal distribution??
Wikipedia says

In probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y

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. $$X|a,b$$??

 

[Jason Swanson]

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 $$X$$??

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 $$y$$.