Let $U_1$, $U_2$ .. $U_n$ be identical and independent random variables distributed Uniform(0, 1). How can I find the cumulative distribution function of the conditional distribution of $U_{(n-1)}$ given $U_{(n)} = c$? Here, $U_{(n-1)}$ refers to the second largest of the aforementioned uniform random variables.
I know that I can find the unconditional distribution of $U_{(n-1)}$: It's just Beta(2, $n - 2 + 1$) or Beta(2, $n-1$) because the ith order statistic of uniforms is distributed Beta(i, $n - i + 1$). However, how do I find the conditional CDF of $U_{(n-1)}$ given that $U_{(n)} = c$??