Page 206 - Probability and Statistical Inference

4. Functions of Random Variables and Sampling Distribution 183

In (4.2.2), the multiplier n! arises because y , ..., y can be arranged among
1
n
themselves in n! ways and the pdf for any such single arrangement amounts
to . Often we are specifically interested in the smallest and largest
order statistics. For the largest order statistic Y , one can find the distribution
n
as follows:

and hence the pdf of Y would be given by
n

in the appropriate space for the Y values. In the same fashion, for the smallest
order statistic Y , we can write:
1

and thus the pdf of Y would be given by
1

in the appropriate space for the Y values.
In the Exercise 4.2.5, we have indicated how one can find the
joint pdf of anY two order statistics Y = X and Y = X .
i n:i j n:j
Using the Exercise 4.2.5, one can derive the joint pdf of Y and Y . In order
n
1
to write down the joint pdf of (Y , Y ) at a point (y , y ) quickly, we adopt the
1
n
1
n
following heuristic approach. Since y , y are assumed fixed, each of the
n
1
remaining n – 2 order statistics can be anywhere between y and y , while
n
1
these could be any n – 2 of the original n random X’s. Now, P{y < X < y } =
1
n
i
F(y ) – F(y ), for each i = 1, ..., n. Hence, with the joint pdf of (Y ,
1
n
1
Y ) would be given by:
n