binomial distribution with random p?

[AtlasFrysmith]

I haven't much training in statistics or math in general, but I was playing around with a signalling model I'm building, mostly for fun, and ran into this issue:

An individual i has a type X(i). X is distributed in some way on the unit interval. Then, a payoff accrues to the individual with probability X.

So, the payoffs for all i have a binomial distribution where p=X, but X is itself a random variable. I haven't chosen the exact distribution of X yet, mostly because I won't even know what to do when I have it. Eventually I want to evaluate the density function of the payoffs, but I don't have a clue how to do that.

Could someone direct me to resources that deal with this sort of problem, if anyone knows of any?


Separately, but as part of the same model, I'm also wondering what happens when a random variable is raised to the power of another random variable. Again, no specific distributions here--the concept itself is eluding me. Any explanations? Thanks.

[jahred]

the way i usually hear of this done is via a beta-binomial mixture distribution. it's a bit advanced, but i'll outline the concept anyway. you're going to want to hunt for a beta-binomial package in R or something to actually evaluate probabilities.

given probability p, X follows a binomial distribution. the beta distribution is continuous and assigns probability to a variable on the unit interval (0,1), so it's a popular choice for modeling probabilities (and can be used to model the probability P in this case).

for
X|P=p ~ bin(n, p)
P ~ beta(a, b),

the density of X, after removing the condition on knowing p, is a tad complex:

f(x) = nCx * beta(a+x, b+n-x) / beta(a, b)

where
- nCx is a binomial coefficient
- beta(a, b) is a beta function, using the form beta(a, b) = gamma(a)gamma(b)/gamma(a + b)
- and gamma(a) is a gamma function

you can use maximum likelihood estimates of a and b (when estimating p), and then your density is simply a function of x. a and b carry all the information about the beta-modeled binomial probability p.

if you have limited exposure to math/stats, i'm aware this might look a bit daunting. but with some experience it's really not too bad, and like i said the applied work could be as easy as finding the correct package in R.

[jahred]

Separately, but as part of the same model, I'm also wondering what happens when a random variable is raised to the power of another random variable. Again, no specific distributions here--the concept itself is eluding me. Any explanations? Thanks.

nothing set in stone happens here, it depends on the random variables themselves. you might wanna google some lecture notes on 'bivariate transformation of random variables', for some background on the general idea.