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For each given $p$, let $X$ have a binomial distribution with parameters $p$ and $N$. Suppose $p$ is distributed according to a beta distribution with parameters r and s. Find the resulting distribution of $X$. When is this distribution uniform on $x=0,1,2,...N$?

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What you want (and much more), is given at Beta-binomial distribution.

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The key concept you need to use is the law of total probability. Let $P$ be the random variable that denotes the success probability of the Binomial distribution. We know the distribution of $P$. Suppose we also know that $P=p$ for some $0 < p < 1$. Given this information, you should be able to write down the probability of the event $\{X=k\}$. Mathematically speaking, we know how to write down $P(X=k \mid P=p)$.

If $P$ were a discrete distribution, the law of total probability will read as $P(X=k) = \sum P(X=k \mid P = p) Pr(P = p)$ where the sum is over all the values taken by the discrete random variable $P$. But in your example, $P$ is continuous which means this summation is replaced by integration.

The final step in finding $P(X=k)$ is to substitute the expressions for $P(X=k \mid P = p)$ and $Pr(P=p)$ into the integral and evaluate the integral. You might find this link useful for evaluating the integral. If you are still getting stuck, show us your work and the difficulties you are facing. I am sure someone will chip in with more hints.