Let $\\X_{(1)},X_{(2)}, \ldots ,X_{(n)}, W $ be an iid sample of distribution with density $f$ which is positive for all $x$. Show that ith $X_{(n)} = \max\{X_{(i)}\}$
$$P(W\le X_{(n)}) =n/(n+1)$$
With an elementary non measure theory approach. How should I go about proving the above claim? I have tried to merge $W$ into the family of order statistic to form a new family with $n+1$ terms though the method doesn't seems to work.
This is not a homework question. It's one that came up in my sample exam paper. Thanks in advance!
Since you mentioned density function, therefore, $P(W<x)=P(W\leq x)$ because the probability of a point in the sample space of a continuous random variable is $0$.
Now, no. of places in between or at the front or back of $X_{(1)},\ldots,X_{(n)}$ where you can place $W$ is $n+1$. Among these places no. of places which will get you $W<X_{(n)}$ is $n$ (not at the back i.e. not after $X_{(n)}$). So, $P(W \leq X_{(n)} = P(W < X_{(n)}) = \frac{n}{n+1}$.