e-Statistics > 4480-5480 Probability and Statistics II

Beta Distribution

When $ X_1,\ldots,X_n$ be iid uniform random variables on $ [0,\theta]$, the pdf of $ X_{(k)}/\theta$ becomes

$\displaystyle f_k(x) = \frac{n!}{(k-1)! (n-k)!} x^{k-1} (1 - x)^{n-k},
\quad 0 \le x \le 1,
$

which is known as the beta distribution with parameters $ \alpha = k =$ and $ \beta = n-k+1 =$ .