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<br /> 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<br /> 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.