Page 63 - Fundamentals of Probability and Statistics for Engineers

46 Fundamentals of Probability and Statistics for Engineers

x
8 Z
f u† du ˆ 0; for x < 0;
<
X
F X x†ˆ 1 3:15†
: ax
1 e ; for x 0.
Let us compute some of the probabilities using f 9x). The probability
X
P (0 < X 1) is numerically equal to the area under f 9x) from x ˆ 0 to
X
x ˆ 1, as shown in Figure 3.6(a). It is given by
Z 1
a
P 0 < X 1†ˆ f x† dx ˆ 1 e :
X
0
The probability P9X > 3) is obtained by computing the area under f 9x) to the
X
right of x ˆ 3. Hence,
1
Z
P X > 3†ˆ f x† dx ˆ e 3a :
X
3
The same probabilities can be obtained from F X 9x) by taking appropriate
differences, giving:
a
a
P 0 < X 1†ˆ F X 1† F X 0†ˆ 1 e † 0 ˆ 1 e ;
P X > 3†ˆ F X 1† F X 3†ˆ 1 1 e 3a †ˆ e 3a :

Let us note that there is no numerical difference between P (0 < X 1) and
P (0 X 1) for continuous random variables, since ( P X ˆ 0) ˆ 0.

3.2.4 MIXED-TYPE DISTRIBUTION

There are situations in which one encounters a random variable that is partially
discrete and partially continuous. The PDF given in Figure 3.7 represents such

F (x)
X

1

x
0
Figure 3.7 A mixed-type probability distribution function, F X 9x)

TLFeBOOK