Let P be a uniform random variable on the interval $(0,1)$ with density function f(p) = 1, $0<p<1$. Let $X_i|P$, i = 1,2,...,n be independent and identically distributed random variables having a Bernoulli distribution with parameter P, where $P(X_i=1|P=p) = p$ and $P(X_i=0|P=p) = 1-p$. Are $X_1,X_2,...,X_n$ exchangeable? Are they independent?
The hint is the problem is a Gamma distribution.