Page 229 - Probability and Statistical Inference

206 4. Functions of Random Variables and Sampling Distribution

(4.4.24) directly as follows: With –∞ < y < ∞, 0 < y < ∞, we get
2
1

With –∞ < y < ∞ and the substitution the marginal pdf
1
g (y ) of Y is then obtained from (4.4.25) as
1
1
1
which coincides with the Cauchy pdf defined in (1.7.31). That is, X /X
1 2
has the Cauchy distribution.

Next, with the substitution
∞, the marginal pdf g (y ) of Y is then obtained from (4.4.25) as
2 2 2

But, note that h(y ) resembles the pdf of the variable for any fixed
1
0 < y < ∞. Hence, must be one for any fixed 0 < y < ∞.
2
2
That is, for all fixed 0 < y < ∞, we get
2

which coincides with the pdf of the x variable. In other words, has the
2
1
distribution. !
The Exercise 4.4.18 provides a substantial generalization in the
sense that X /X can be claimed to have a Cauchy distribution
1
2
even when (X , X ) has the N (0, 0, σ , σ , ρ) distribution. The
2
2
2
2
1
transformation used in the Exercise 4.4.18 is also different
from the one given in the Example 4.4.16.
4.5 Special Sampling Distributions
In the Section 1.7, we had listed some of the standard distributions by
including their pmf or pdf. We recall some of the continuous distributions,