1.7. Continuous Random Variables 51

Taking the derivative ofFX(x), we obtain the pdf ofX:

fX(x)=

{

2 x 0 ≤x< 1

0elsewhere.

(1.7.6)

For illustration, the probability that the selected point falls in the ring with radii
1 /4and1/2isgivenby

P

(

1

4

<X≤

1

2

)

=

∫ (^12)
(^14)
2 wdw=w^2
∣
∣
∣
∣
∣
(^12)
(^14)

3
16
.
Example 1.7.2.Let the random variable be the time in seconds between incoming
telephone calls at a busy switchboard. Suppose that a reasonable probability model
forXis given by the pdf
fX(x)=
{ 1
4 e
−x/ (^40) <x<∞
0elsewhere.
Note thatfXsatisfies the two properties of a pdf, namely, (i)f(x)≥0 and (ii)
∫∞
0
1
4
e−x/^4 dx=−e−x/^4
∣
∣
∣
∣
∞
0
=1.
For illustration, the probability that the time between successive phone calls exceeds
4 seconds is given by
P(X>4) =
∫∞
4
1
4
e−x/^4 dx=e−^1 =0. 3679.
The pdf and the probability of interest are depicted in Figure 1.7.1. From the figure,
the pdf has a long right tail and no left tail. We say that this distribution isskewed
rightor positively skewed. This is an example of a gamma distribution which is
discussed in detail in Chapter 3.

1.7.1 Quantiles

Quantiles (percentiles) are easily interpretable characteristics of a distribution.

Definition 1.7.2(Quantile). Let 0 <p< 1 .Thequantileof orderpof the

distribution of a random variableX is a valueξpsuch thatP(X<ξp)≤pand

P(X≤ξp)≥p.Itisalsoknownasthe(100p)thpercentileofX.

Examples include themedianwhich is the quantileξ 1 / 2. The median is also

called the second quartile. It is a point in the domain ofXthat divides the mass

of the pdf into its lower and upper halves. The first and third quartiles divide each

of these halves into quarters. They are, respectivelyξ 1 / 4 andξ 3 / 4. We label these

quartiles asq 1 ,q 2 andq 3 , respectively. The difference iq =q 3 −q 1 is called the