Distribution of random variable for which density parameter is another random variable

Question

Say $X$ is a random variable that has a density $f_p$ depending on a parameter $p$ (for instance the parameter for a normal random variable would be the mean an variance or for a Bernoulli the probability of success, ect...). Now say that $p$ is also a random variable in the problem, call it $P$. Now given a set $A$, how can I express $\mathbb{P}(X\in A)$ depending on the the density $f_p$ and also using the distribution of $P$ (call it $\mu_P$)? I think this has maybe something to do with mixture models.

Answer 1

What you are describing is called a compound distribution.

Given a set $A$ we can get $P(X \in A)$ by integrating over the conditional density. Let $P \sim g(p)$:

$$P(X \in A) = \int_{p \in \Omega_p}\int_A g(p)f_p(x)dxdp$$

Where $\Omega_p$ is just the range of $P$.

Answer 2

You have to be precise about what you mean by $X$ has a density that depends on random parameter $P$. You could formulate this mathematically as asserting that the conditional distribution of $X$ given $P = p$ has density $f_p$. This means you want a joint distribution $(P, X)$ where $P \sim \mu_P$ and $P(X \in dx \mid P = p) = f_p(x)\,dx$. Such a joint distribution exists, and can be defined in a natural way by $$P((P, X) \in S) = \int \mu_P(dp) \int f_p(x)\,dx \, 1_{S}(p,x).$$ You can define arbitrary joint distributions with desired conditional distributions in this way, as a composition of transition kernels.