Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

54 Probability and Distributions

0.05

0.10

− 8 q 1 q 2 q 3

x

f(x)

Figure 1.7.2:A graph of the pdf (1.7.9) showing the three quartiles,q 1 ,q 2 ,and
q 3 , of the distribution. The probability mass in each of the four sections is 1/4.

It follows that the pdf ofYis

fY(y)=

{

10 <y< 1

0elsewhere.

Example 1.7.5.LetfX(x)=^12 ,− 1 <x<1, zero elsewhere, be the pdf of a
random variableX.NotethatXhas a uniform distribution with the interval of
support (− 1 ,1). Define the random variableY byY =X^2. We wish to find the
pdf ofY.Ify≥0, the probabilityP(Y≤y)isequivalentto

P(X^2 ≤y)=P(−

√

y≤X≤

√

y).

Accordingly, the cdf ofY,FY(y)=P(Y≤y), is given by

FY(y)=

⎧

⎪⎨

⎪⎩

0 y< 0

∫√y

−√y

1

2 dx=

√

y 0 ≤y< 1

11 ≤y.

Hence, the pdf ofYis given by

fY(y)=

{ 1

2 √y^0 <y<^1

0elsewhere.

These examples illustrate thecumulative distribution function technique.

The transformation in Example 1.7.4 is one-to-one, and in such cases we can obtain