# Beta-binomial distribution for scaled and translated Beta

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 $$L the beta integral transforms to:" $$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 $$p_{BB(n, \alpha, \beta)}$$.

Cross Post:
This is a cross post from Cross Validated SE. I am aware that doing this is suboptimal but decided that it might be appropriate in this case.