Question.

Let be a random sample from uniform distribution , where is an unknown parameter. Let be the k-th smallest value among for .

(a) find density function of

(b) Find .

You may use the identity for integers i,j>0.

Answer.

(a) can be found in a textbook so I'll skip it here.

(b)

I will now integrate not y but which will (1) change the upper bound of integration to 1 and (2) so I need to add one more theta into the integrand. I then take all non-y items outside the integral, and apply the 'identity' given above to what's left under the integral sign

Now, does it make sense as an expected value of k-th order statistic? It is between 0 and theta (since k/(n+1) is between 0 and 1). But I cannot say anything else, I cannot visualise a distribution where any value of y between 0 and theta has equal probability to happen.