Introduction to Probability and Statistics for Engineers and Scientists

94 Chapter 4:Random Variables and Expectation

In words, Equation 4.2.1 states that the probability thatXwill be inBmay be obtained
by integrating the probability density function over the setB. SinceXmust assume some
value,f(x) must satisfy

1 =P{X∈(−∞,∞)}=

∫∞

−∞

f(x)dx

All probability statements aboutXcan be answered in terms off(x). For instance, letting
B=[a,b], we obtain from Equation 4.2.1 that

P{a≤X≤b}=

∫b

a

f(x)dx (4.2.2)

If we leta=bin the above, then

P{X=a}=

∫a

a

f(x)dx= 0

In words, this equation states that the probability that a continuous random variable will
assume anyparticularvalue is zero. (See Figure 4.3.)
The relationship between the cumulative distributionF(·) and the probability density
f(·) is expressed by

F(a)=P{X∈(−∞,a]} =

∫a

−∞

f(x)dx

Differentiating both sides yields

d

da

F(a)=f(a)

a

1

Area of shaded region = P { a < X < b }

x

f(x) = e−x

b

FIGURE 4.3 The probability density function f(x)=

{

e−x x≥ 0

0 x< 0

.