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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.
Take any n tuple $x$ of 0's and 1's with k 1's and n-k 0's. Then $$ P((X_1,\ldots, X_n) = x|P=p) = p^k(1-p)^{n-k} $$ So, $$P((X_1,\ldots, X_n) = x) = \int_0^1p^k(1-p)^{n-k}dp = \frac{\Gamma(k+1)\Gamma(n-k+1)}{\Gamma(n+2)}$$
Since the above expression only depends upon the number of $1$'s, you can permute them and obtain the same probability. This shows exchangeability.
But $P(X_1 = 1) = P(X_2 = 1) = \int_0^1pdp = 1/2$. But $$P(X_1=1,X_2=1) = \frac{\Gamma(3)}{\Gamma(4)} = 1/3$$
so they are not independent!
