Continuous Random Variables and PDFs
Cumulative Distribution Functions
Normal Random Variables
Multiple Continuous Random Variables
Conditioning
The Continuous Bayes’ Rule
.
.
EE 230
General Random Variables
Department of Electrical and Electronics Engineering
Middle East Technical University
April 21, 2016
1/75

Continuous Random Variables and PDFs
Cumulative Distribution Functions
Normal Random Variables
Multiple Continuous Random Variables
Conditioning
The Continuous Bayes’ Rule
.
General Random Variables
Random variables with a
continuous range of
possible values
are quite common, e.g., thermal
noise current intrinsic to all resistors (due to heating
of a resistor), your exact weight,
....
In fact, some
discrete random variables are obtained through some
operation on continuous ranges.
The use of continuous models may result in insights
that would not be possible with discrete modeling.
All of the concepts and tools introduced for discrete
random variables have continuous counterparts.
2/75

Continuous Random Variables and PDFs
Cumulative Distribution Functions
Normal Random Variables
Multiple Continuous Random Variables
Conditioning
The Continuous Bayes’ Rule
Properties of PDF
Some Continuous Random Variables and Their PDFs
Expectation
.
Continuous Random Variables and PDFs
.
Example
.
.
Angle of an arbitrarily drawn line. Consider that all
outcomes are equally likely. What is the probability that
the angle will be less than
π
?
3/75

Continuous Random Variables and PDFs
Cumulative Distribution Functions
Normal Random Variables
Multiple Continuous Random Variables
Conditioning
The Continuous Bayes’ Rule
Properties of PDF
Some Continuous Random Variables and Their PDFs
Expectation
.
Continuous Random Variables and PDFs
.
Definition
.
.
A random variable
X
is
continuous
if there is a
nonnegative func.
f
X
, called the
probability density
function (PDF)
of
X
, such that
P
(
X
∈
B
) =
B
f
X
(
x
)
dx
,
for
every subset
B
of the real line.
In particular,
P
(
a
≤
X
≤
b
) =
b
a
f
X
(
x
)
dx
4/75

Continuous Random Variables and PDFs
Cumulative Distribution Functions
Normal Random Variables
Multiple Continuous Random Variables
Conditioning
The Continuous Bayes’ Rule
Properties of PDF
Some Continuous Random Variables and Their PDFs
Expectation
.
Continuous Random Variables and PDFs
.
Definition
.
.
A random variable
X
is
continuous
if there is a
nonnegative func.
f
X
, called the
probability density
function (PDF)
of
X
, such that
P
(
X
∈
B
) =
B
f
X
(
x
)
dx
,
for
every subset
B
of the real line.
In particular,
P
(
a
≤
X
≤
b
) =
b
a
f
X
(
x
)
dx
(area under the PDF)
4/75

Continuous Random Variables and PDFs
Cumulative Distribution Functions
Normal Random Variables
Multiple Continuous Random Variables
Conditioning
The Continuous Bayes’ Rule
Properties of PDF
Some Continuous Random Variables and Their PDFs
Expectation
.

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