Page 57 - Fundamentals of Probability and Statistics for Engineers

40 Fundamentals of Probability and Statistics for Engineers
. It exists for discrete and continuous random variables and has values between
0and 1.
. It is a nonnegative, continuous-to-the-left, and nondecreasing function of the
real variable x. Moreover, we have
F X 1† ˆ 0; and F X ‡1† ˆ 1: 3:2†
. If a and b are two real numbers such that a < b , then
P a < X b†ˆ F X b† F X a†: 3:3†

This relation is a direct result of the identity

P X b†ˆ P X a†‡ P a < X b†:
We see from Equation (3.3) that the probability of X having a value in an
arbitrary interval can be represented by the difference between two values of
the PDF. Generalizing, probabilities associated with any sets of intervals are
derivable from the PDF.
Example 3.1. Let a discrete random variable X assume values 1, 1, 2, and 3,
1 1 1
1
with probabilities , , ,and , respectively. We then have
4 8 8 2
0; for x < 1;
8
>
>
1
>
>
> ; for 1 x < 1;
>
> 4
>
< 3
F X x†ˆ ; for 1 x < 2;
8
>
1
>
>
>
> ; for 2 x < 3;
>
> 2
>
:
1; for x 3:
This function is plotted in Figure 3.1. It is typical of PDFs associated with
discrete random variables, increasing from 0 to 1 in a ‘staircase’ fashion.
A continuous random variable assumes a nonenumerable number of values
over the real line. Hence, the probability of a continuous random variable
assuming any particular value is zero and therefore no discrete jumps are
possible for its PDF. A typical PDF for continuous random variables is
shown in Figure 3.2. It has no jumps or discontinuities as in the case of the
discrete random variable. The probability of X having a value in a given
interval is found by using Equation (3.3), and it makes sense to speak only of
this kind of probability for continuous random variables. For example, in
Figure 3.2.
P 1 < X 1†ˆ F X 1† F X 1†ˆ 0:8 0:4 ˆ 0:4:
Clearly, P9X ˆ a) ˆ 0 for any . a

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