Page 61 - Fundamentals of Probability and Statistics for Engineers

44 Fundamentals of Probability and Statistics for Engineers
3.2.3 PROBABILITY DENSITY FUNCTION FOR CONTINUOUS
RANDOM VARIABLES

X
For a continuous random variable , its PDF, F X 9x), is a continuous function
of x, and the derivative

dF X x†
f X x†ˆ ; 3:10†
dx

exists for all x. The function f 9 x) is called the probability density function (pdf),
X
or simply the density function , of X . (1)
Since F X 9x) is monotone nondecreasing, we clearly have

f x† 0 for all x: 3:11†
X
Additional properties of f 9x) can be derived easily from Equation (3.10);
X
these include

x
Z
F X x†ˆ f u† du; 3:12†
X
1
and
Z 1 9
f x† dx ˆ 1; >
X >
1 =
b 3:13†
Z
>
P a < X b†ˆ F X b† F X a†ˆ f x† dx: >
X
;
a
An example of pdfs has the shape shown in Figure 3.5. As indicated by
Equations (3.13), the total area under the curve is unity and the shaded area
from a to b gives the probability P9a < X b). We again observe that the
knowledge of either pdf or PDF completely characterizes a continuous random
variable. The pdf does not exist for a discrete random variable since its
associated PDF has discrete jumps and is not differentiable at these points of
discontinuity.
Using the mass distribution analogy, the pdf of a continuous random variable
plays exactly the same role as the pmf of a discrete random variable. The

1
Note the use of upper-case and lower-case letters, PDF and pdf, to represent the distribution and
density functions, respectively.

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