Page 207 - Probability and Statistical Inference

184 4. Functions of Random Variables and Sampling Distribution

In the same fashion, one can easily write down the joint pdf of any subset
of the order statistics. Now, let us look at some examples.

In the Exercises 4.2.7-4.2.8, we show how one can find the pdf
of the range, X – X when one has the random samples
n:1
n:n
X , ..., X from the Uniform distribution on the interval (0, θ) with
1 n
θ ∈ ℜ or on the interval (θ, θ + 1) with θ ∈ ℜ respectivelY.
+
Example 4.2.7 Suppose that X , ..., X are iid random variables distributed
1
n
as Uniform on the interval (0, θ) with θ > 0. Consider the largest and smallest
order statistics Y and Y respectively. Note that
n 1

and in view of (4.2.4) and (4.2.6), the marginal pdf of Y and Y will be
n
1
respectively given by g(y) = ny θ and h(y) = n(θ – y) θ for 0 < y < θ. In
n–1 –n
n–1 –n
view of (4.2.7), the joint pdf of (Y , Y ) would be given by f(y , y ) = n(n –
n
1
n
1
1)(y – y ) θ for 0 < y < y < θ. !
n–2 –n
n 1 1 n
The next example points out the modifications needed in (4.2.4)
and (4.2.6)-(4.2.7) in order to find the distributions of various
order statistics from a set of independent, but not identically
distributed continuous random variables.
Example 4.2.8 Consider independent random variables X and X where
2
1
their respective pdf’s are given by f (x) = 1/3x I(–1 < x < 2) and
2
1
f (x) = 5/33x I(–1 < x < 2). Recall that here and elsewhere I(.) stands
4
2
for the indicator function of (.). One has the distribution functions
for
–1 < x < 2. Hence, for –1 < y < 2, the distribution function F(y) of Y = X ,
2:2
2
the larger order statistic, can be found as follows:
since the X’s are independent. By differentiating F(y) with respect to y, one
can immediately find the pdf of X . Similarly one can handle the distribution
2:2
of X , the smaller order statistic. The same idea easily extends for n indepen-
2:1
dent, but not identically distributed continuous random variables. !