Let $U_1, ..., U_n$ be independent uniform random variables, and let $V$ be uniform and independent of the $U_i$.

Question

Let $U_1, ..., U_n$ be independent uniform random variables, and let $V$ be uniform and independent of the $U_i$.

a) Find $P(V<U_{(n)})$,

b) Find $P(U_{(1)}<V<U_{(n)})$

I found that the cumulative density function of the largest order statistics $U_{(n)}$ is given by $F_n(u)=u^n$ for $0<u<1$. How do I use this then to find parts a and b?

Answer 1

$P(V\lt U_{(n)})=E[\mathbf{1}_{V\lt U_{(n)}}]=E\left[E[\mathbf{1}_{V\lt U_{(n)}}|U_{(n)}]\right]=\dots$

Can you go on from there?

For part b), I'd use the same method, conditioning by $U_{(n)}-U_{(1)}$ (you need to determine the distribution of this random variable first).

Answer 2

[The following reasoning works for any iid sequence with continuous marginal distribution.]

Think of $V$ as $X_{n+1}$. The event $\{V>U_{(n)}\}$ then coincides with the event that the largest of $X_1,X_2,\ldots,X_{n+1}$ is $X_{n+1}$. By symmetry, the probability of this event is $1/(n+1)$. Therefore $$ P(V<U_{(n)})=P(V\le U_{(n)})=1-P(V>U_{(n)})=1-1/(n+1)=n/(n+1). $$ Symmetrical reasoning leads to $P(V\le U_{(1)})=P(V<U_{(1)})=1/(n+1)$, and the answer to (b) is now a short step away.