Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.5. Random Variables 41

F(x)

x

1

1

(0, 0)

Figure 1.5.2:Distribution function for Example 1.5.4.

The cdfs displayed in Figures 1.5.1 and 1.5.2 show increasing functions with lower
limits 0 and upper limits 1. In both figures, the cdfs are at least right continuous.
As the next theorem proves, these properties are true in general for cdfs.

Theorem 1.5.1.LetXbe a random variable with cumulative distribution function
F(x).Then

(a) For allaandb,ifa<b,thenF(a)≤F(b)(Fis nondecreasing).

(b)limx→−∞F(x)=0(the lower limit ofFis 0).

(c)limx→∞F(x)=1(the upper limit ofF is 1).

(d)limx↓x 0 F(x)=F(x 0 )(Fis right continuous).

Proof:We prove parts (a) and (d) and leave parts (b) and (c) for Exercise 1.5.10.
Part (a): Becausea<b,wehave{X≤a}⊂{X≤b}. The result then follows
from the monotonicity ofP; see Theorem 1.3.3.
Part (d): Let{xn}be any sequence of real numbers such thatxn↓x 0 .LetCn=
{X≤xn}. Then the sequence of sets{Cn}is decreasing and∩∞n=1Cn={X≤x 0 }.
Hence, by Theorem 1.3.6,

lim

n→∞

F(xn)=P

(∞

⋂

n=1

Cn

)

=F(x 0 ),

which is the desired result.

The next theorem is helpful in evaluating probabilities using cdfs.

Theorem 1.5.2. LetXbe a random variable with the cdfFX. Then fora<b,
P[a<X≤b]=FX(b)−FX(a).