Recall, that a binomial distribution in which the probability of success at each trial is randomly drawn from a beta distribution results in the so called beta-binomial distribution. One can calculate the probability mass function explicitly as

$$p_{BB(n, \alpha, \beta)}(k) := \mathbb{P}[X=k \mid n, \alpha, \beta] = \binom{n}{k} \frac{B(\alpha + k, \beta + n-k)}{B(\alpha, \beta)} \; .$$

Now, following up this answer I find myself in the following situation:

Given $n$ Bernoulli trials, such that the success rate for each trial is modeled with a distribution $p \sim(U-L)X_B + L$, where $X_B\sim Beta(\alpha, \beta)$ is beta distributed and $0 \leq L < U \leq 1$ are constants. What is the corresponding probability mass function for this modified beta-binomial distribution?

probabilityislogic mentioned in his post that

"[...] noting that making the change of variables $p=\frac{x-L}{U-L}$ for $L<x<U$ the beta integral transforms to:" $$B(\alpha,\beta)=\int_{L}^{U}\frac{(x-L)^{\alpha-1}(U-x)^{\beta-1}}{(U-L)^{\alpha+\beta-1}}dx$$

But I'm not entirely sure how to use that information to get the desired pmf. More specifically, I wasn't able to deduce an expression of $p_{BB(L, U, n, \alpha, \beta)}$ in terms of $p_{BB(n, \alpha, \beta)}$.


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