Order Statistics (Sample Median, Range)

Question

I need your help on how to start solving this problem. I really have a hard time and do not know where to start.

Let X1, X2 , … , Xn be a random sample from U(0, a) and let (1), (2), … , () denote the order statistics. The range is defined as = () − (1); the midrange, a measure of location like the sample median, is defined as = 1/2 (X(n) + X(1).

a. Derive the joint pdf of and .

b. Derive the sampling distribution of R.

c. Suppose = 1 and = 2. Find the probability that the two observations will not differ by less than 0.5.

Answer 1

HINT

Start with thinking about / researching some facts about order statistics of iid uniform RVs. This one may be useful: conditional on the maximum $X(n) = m,$ the remaining order statistics are distributed the same as the order statistics of a sample of $n-1$ independent $U(0,m)$ variables.

Answer 2

Sketch of a proof:

1) Prove that given $X_{(n)}=t$, the other $n-1$ random variable are distributed $\operatorname{U}(0,t)$, for $0<t<a$

2) Using first part, find the conditional probability $f_{X_{(1)}|X_{(n)}}(x|t)$, for $0<x<t<a$

3) Find the joint pdf of $X_{(1)}$ and $X_{(n)}$: $f_{X_{(1)},X_{(n)}}(x,t)=f_{X_{(1)}|X_{(n)}}(x|t)$ $f_{X_{(n)}}(t)$

4) Find the joint pdf of $R$ and $V$ by performing the transformation of random variables which you stated (and paying attention to the support of the joint pdf)