Assume that $Q$ is a random variable with density proportional to $q$ for $0 < q < 1$. Given $Q = q$, $N$ has a binomial distribution with parameters $n$ and $q$.
What is the probability mass function of $N$?
2026-03-27 10:30:07.1774607407
How to find binomial pmf with probability = another random variable
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If $Q$ has density $f$ on $[0,1]$ and if, conditionally on $Q=q$, $N$ has conditional probability mass function $p(\ \mid q)$, then Bayes formula yields $$ P(N=k)=\int_0^1P(N=k\mid Q=q)\,f(q)\,\mathrm dq=\int_0^1p(k\mid q)\,f(q)\,\mathrm dq. $$ Can you finish?