PDF of difference of Beta$(1, n)$ and Beta$(n, 1) =$ Beta(n - 1, 2)

Question

Given $X_1 \sim \text{Beta}(1, n)$ and $X_2 \sim \text{Beta}(n, 1)$, how do I show that $X_2 - X_1 \sim \text{Beta}(n - 1, 2)$?

I have seen a similar question, but that was not particularly helpful.

Answer 1

Short answer. The distribution of $Y = X_2 - X_1$ is not going to be $Beta(n-1, 2)$. For starters the support of the Beta distribution is in the interval (0,1), and $Y < 0$ with nonzero probability.

This may be true, however, when $n \rightarrow \infty$. Here is a quick simulation which illustrates this point. Note that even in the bottom panel, where $n=10$, the minimum of the simulated $Y$ values is $-0.1$ illustrating that this is NOT a beta distribution.

You are somewhat misunderstanding the link you refer to in your comment. If $X_1$ and $X_2$ are independent, and have the distributions you state, then they will not be independent (see above figures).

If however, $U_1, \cdots U_n$ are iid $U(0,1)$ observations, and we let $X_2 = \max(U_i)$ and $X_1 = \min(U_i)$ then marginally the distributions of $X_1$ and $X_2$ will be what you state. In this case, $X_2 - X_1$ will follow the desired distribution, but they key point is that they are not independent. Namely, $X_2$ must be at least as big as $X_1$, so that the difference will always be between $0$ and $1$.