1.5. Random Variables 39

there are some intuitive probabilities. For instance, because the number is chosen

at random, it is reasonable to assign

PX[(a, b)] =b−a,for0<a<b< 1. (1.5.3)

It follows that the pdf ofXis

fX(x)=

{

10 <x< 1

0elsewhere.

(1.5.4)

For example, the probability thatX is less than an eighth or greater than seven
eighths is

P

[{

X<

1

8

}

∪

{

X>

7

8

}]

=

∫ (^18)
0
dx+
∫ 1
(^78)
dx=
1
4
.
Notice that a discrete probability model is not a possibility for this experiment. For
any pointa,0<a<1, we can choosen 0 so large such that 0<a−n− 01 <a<
a+n− 01 <1, i.e.,{a}⊂(a−n− 01 ,a+n− 01 ). Hence,
P(X=a)≤P
(
a−
1
n
<X<a+
1
n
)

2
n
, for alln≥n 0. (1.5.5)
Since 2/n→0asn→∞andais arbitrary, we conclude thatP(X=a)=0for
alla∈(0,1). Hence, the reasonable pdf, (1.5.4), for this model excludes a discrete
probability model.
Remark 1.5.1.In equations (1.5.1) and (1.5.2), the subscriptXonpXandfX
identifies the pmf and pdf, respectively, with the random variable. We often use
this notation, especially when there are several random variables in the discussion.
On the other hand, if the identity of the random variable is clear, then we often
suppress the subscripts.
The pmf of a discrete random variable and the pdf of a continuous random
variable are quite different entities. The distribution function, though, uniquely
determines the probability distribution of a random variable. It is defined by:
Definition 1.5.2(Cumulative Distribution Function).LetXbe a random variable.
Then itscumulative distribution function(cdf)isdefinedbyFX(x),where
FX(x)=PX((−∞,x]) =P({c∈C:X(c)≤x}). (1.5.6)
As above, we shortenP({c∈C: X(c)≤x})toP(X ≤x).Also,FX(x)is
often called simply the distribution function (df). However, in this text, we use the
modifiercumulativeasFX(x) accumulates the probabilities less than or equal tox.
The next example discusses a cdf for a discrete random variable.
Example 1.5.3. Suppose we roll a fair die with the numbers 1 through 6 on it.
LetXbe the upface of the roll. Then the space ofXis{ 1 , 2 ,..., 6 }and its pmf
ispX(i)=1/6, fori=1, 2 ,...,6. Ifx<1, thenFX(x)=0. If1≤x<2, then
FX(x)=1/6. Continuing this way, we see that the cdf ofXis an increasing step
function which steps up bypX(i)ateachiin the space ofX. The graph ofFXis
given by Figure 1.5.1. Note that if we are given the cdf, then we can determine the
pmf ofX.