STAT 230: Introduction to Probability and
Random Variables
Summer 2009
Chapter 3: Random Variables of the Continuous Type
1
PDF and CDF
Thus far, we have only discussed discrete random variables in detail.
Recall
that discrete random variables describe things that can be counted (number of
successes in a sequence of Bernoulli trials, etc). There is another class of random
variables that describe things that can be measured but not counted. Examples
are
•
height, weight, length, area, volume, etc.
•
lifetimes, waiting times, etc.
These continuous random variables can take values in an interval, unlike discrete
random variables that take values in a countable set, like 1, 2, 3,
. . ..
Recall
that with a discrete random variable
X
we had a PMF
p
X
(
x
)
≥
0 and all prob
abilities are found by summing this function over suitable values of
x
. Things
are basically the same for a continuous random variable
X
: it has a function
f
X
(
x
)
≥
0, called the PDF and all probabilities are found by integrating over
suitable values of
x
.
Definition 1.
Probability density function (PDF)
A random variable X is called continuous if there is a function
f
X
(
x
)
, called
the probability density function (PDF), that satisfies the following properties:
(i)
f
X
(
x
)
≥
0
for all
x
.
(ii)
R
∞
∞
f
X
(
x
)
dx
= 1
.
(iii) For any constants
a < b
, the probability that
X
is between
a
and
b
is:
P
(
a
≤
X
≤
b
) =
R
b
a
f
X
(
x
)
dx
1
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that is,
P
(
a
≤
X
≤
b
)
is the area under the curve
f
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 Fall '08
 unknown
 Probability theory, probability density function, cdf FX

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