52 Probability and Distributions

f(x)

x

2 4

0.2

0.1

(0, 0)

Figure 1.7.1:In Example 1.7.2, the area under the pdf to the right of 4 isP(X>

4).

interquartile rangeofX. The median is often used as a measure of center of the
distribution ofX, while the interquartile range is used as a measure ofspreador
dispersionof the distribution ofX.
Quantiles need not be unique even for continuous random variables with pdfs.
For example, any point in the interval (2,3) serves as a median for the following
pdf:

f(x)=

⎧

⎨

⎩

3(1−x)(x−2) 1<x< 2

3(3−x)(x−4) 3<x< 4

0elsewhere.

(1.7.7)

If, however, a quantile, sayξp, is in the support of an absolutely continuous random
variableXwith cdfFX(x)thenξpis the unique solution to the equation:

ξp=FX−^1 (p), (1.7.8)

whereFX−^1 (u) is the inverse function ofFX(x). The next example serves as an
illustration.

Example 1.7.3.LetXbe a continuous random variable with pdf

f(x)=

ex

(1 + 5ex)^1.^2

, −∞<x<∞. (1.7.9)

This pdf is a member of the logF-family of ditributions which is often used in the

modeling of the log of lifetime data. Note thatXhas the support space (−∞,∞).

The cdf ofXis

F(x)=1−(1 + 5e−x)−.^2 , −∞<x<∞,