Page 205 - Probability and Statistical Inference

182 4. Functions of Random Variables and Sampling Distribution

That is, W has a uniform distribution on the interval (0, 1). Next, define
U = – log (F(X)) and V = –2log(F(X)). The domain space of U and V are both
(0, ∞). Now, for 0 < u < ∞, one has the df of U,

so that U has the pdf f(u) = G(u) = e , which corresponds to the pdf of a
–u
standard exponential random variable which is defined as Gamma(1, 1). Re-
fer to (1.7.24). Similarly, for 0 < v < ∞, we write the df of V,

so that V has the pdf g(v) = H(v) = ½e –1/2v , which matches with the pdf of
a Chi-square random variable with two degrees of freedom. Again refer back
to Section 1.7 as needed. !
Example 4.2.6 Suppose that Z has a standard normal distribution and let Y
= Z . Denote ∅(.) and Φ (.) respectively for the pdf and df of Z. Then, for 0
2
< y < ∞, we have the df of Y,

and hence the pdf of Y will be given by

Now g(y) matches with the pdf of a Chi-square random variable with one
degree of freedom. That is, Z is distributed as a Chi-square with one degree
2
of freedom. Again, refer back to Section 1.7. !
Suppose that X has a continuous random variable with its df
F(x). Then, F(X) is Uniform on the interval (0, 1), –log{F(X)}
is standard exponential, and –2log{F(X)} is

4.2.3 The Order Statistics

Let us next turn to the distributions of order-statistics. Consider independent
and identically distributed (iid) continuous random variables X , ..., X having
n
1
the common pdf f(x) and the df F(x). Suppose that X ≤ X ≤ ... ≤ X stand
n:1
n:n
n:2
for the corresponding ordered random variables where X is referred to as
n:i
th
the i order statistic. Let us denote Y = X for i = 1, ..., n and Y = (y , ..., y ).
n:i
i
n
1
The joint pdf of Y , ..., Y , denoted by f (y , ..., y ), is given by
1 n Y 1 n