Chapter 2: Random Variables
2.1. Concept of a Random Variable
2.2. Distribution Functions
2.3. Density Functions
Functions of random variables
Hypergeometric Distribution
2.4. Mean Values and Moments
2.5. The Gaussian Random Variable
2.6. Density Functions Related to Gaussian
2.7. Other Probability Density Functions
2.8. Conditional Probability Distribution and Density Functions
2.9. Examples and Applications
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
Conditional Probability (Review from Chapter 1)
Defining the conditional probability of event A given that event B has occurred.
Using a Venn diagram, we know that B has occurred … then the probability that A has occurred
given B must relate to the area of the intersection of A and B …
Pr  A  B   Pr  A | B   Pr B  , for PrB   0
or
Pr  A | B  
Pr  A  B 
, for PrB   0
Pr B 
For elementary events,
Pr  A | B  
Pr  A  B  Pr  A, B 

, for PrB   0
Pr B 
Pr B 
If A is a subset of B, then the conditional probability must be
Pr  A | B  
Pr  A  B  Pr  A

, for A  B
Pr B 
Pr B 
Therefore, it can be said that
Pr  A | B  
Pr  A  B  Pr  A

 Pr  A , for A  B
Pr B 
Pr B 
If B is a subset of A, then the conditional probability becomes
Pr  A | B  
Pr  A  B  Pr B 

 1 , for B  A
Pr B 
Pr B 
If A and B are mutually exclusive,
Pr  A | B  
Pr  A  B 
0

 0 , for B  A  
Pr B 
Pr B 
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
2 of 15
ECE 3800
Conditional Probability Distribution and Density Functions
Using the Probability Distribution Function (PDF), define
F  x | M   Pr  X  x | M 
Pr  X  x | M  
Pr  X  x, M 
, for Pr M   0
Pr M 
where  X  x, M  is the event of all outcomes  such that
X    x and   M
Note: X   is the value of the random variable when the experimental outcome is  .
Every M may be an event that depends upon some other random variable, which may be either
continuous or discrete. The examples to be considered are in chapter 3.
It can be shown that F x | M  is a valid probability distribution function with all the expected
characteristics:
1. 0  F  x | M   1, for    x  
2. F   | M   0 and F  | M   1
3. F x | M  is non-decreasing as x increases
4. Pr  x1  X  x2 | M   F x2 | M   F  x1 | M 
The event M that conditions the probability has several possibilities:
1. Every M may be an event that can be expressed in terms of the random variable X.
The case that is discussed in chapter 2 and that we start with,
2. Every M may be an event that depends upon some other random variable, which may be
either continuous or discrete. The examples to be considered in chapter 3.
3. Every M may be an event that depends upon both the random variable X and some other
random variable. Examples and cases not dealt with in this text.
For our purposes (#1 above) envision that
M  X  m or M  X  m
So now we need to investigate the two cases of interest completely …
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
3 of 15
ECE 3800
Using M  X  m then
F  x | M   Pr  X  x | X  m  
Pr  X  x, X  m 
Pr  X  m 
There are two possibilities depending on the values of x and m selected.
If x  m , then the event described by M  X  m is contained in the event described for
X  x and
Pr  X  x, X  m  Pr  X  m 
F x | M  

 1 , for x  m
Pr  X  m 
Pr  X  m 
Since the event M is a subset and has occurred, then we know that x will occur based on M.
If x  m , then the event described by M  X  m contains the event described for X  x
and
Pr  X  x, X  m  Pr  X  x  Pr  X  x 
, for x  m
F x | M  


Pr  X  m 
Pr  X  m  Pr  X  m 
The distribution function is effectively scaled by the probability of the event. As a result the new
probability density function is a scaled version of the initial distribution, but a ceiling function is
applied for those points beyond m that would be greater than 1.0.
A new probability density function may result from the above and it can be defined as
f x | M  
dF  x | M 
dx
Properties of the new pdf must include
1.
f  x | M   0, for    x  

2.
 f x | M   dx  1

3. F  x | M  
x
 f u | M   du

4. Pr  x1  X  x 2  
x2
 f x | M   dx
x1
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
4 of 15
ECE 3800
Note that for the conditional pdf, where x  m , we must have
f x | M  
dF  x | M  d 1

 0 , for x  m
dx
dx
and
f x | M  
dF  x | M 
1
d Pr  X  x  f  x 



, for x  m
dx
Pr  X  m 
dx
F m 
Summary:
For x  m
F x | M  
Pr  X  x, X  m  Pr  X  m 

1
Pr  X  m 
Pr  X  m 
For x  m
F x | M  
Pr  X  x, X  m  Pr  X  x  F  x 


Pr  X  m 
Pr  X  m  F m 
 f x 
, xm

f x | M    F m 
0,
xm
All normal operations can be performed on the new density and distribution functions.
Moments
The moments of a random variable are defined as the expected value of the powers of the
measured output or …


n
 x
E X |M 
n
 f  x | M   dx

The expected value of a function is:
E g  X  | M  

 g  X   f x | M   dx

Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
5 of 15
ECE 3800
Example p. 100: Gaussian density function given that the event, M, is less
than or equal to the mean value
f X x  


 x X 2 
, for    x  
 exp


2
2  
 2 

1
X is the mean and  is the variance
where
For the event M,
F x  M  X   1
2
Then, constructing a conditional probability distribution, F  x  X | M  , the density function is
defined as
 x X 2 
f x 
f x 
2
 , for    x  X


 exp
f x | M  


2
1
F X
2  
 2 

2

 
and

f x | M   0 , for x  X
The new conditional distribution and density has a mean and variance that can be computed.
The new mean is
X
 x X 2 
2
  dx
E x    x 
 exp
2


2  
x  
 2 


E x  
 u  X 
0
x  
E x   2  X 
0

x  
From math tables
  u2
 exp
2
2  
 2 
1


  u2
 exp
2
2  
 2 
2

  du

0

  u2
2
  du 
  u  exp
2
2   x  

 2 

  du

2 
  x2 

  dx  exp  x 
 exp
 2  2 
 2  2 
2  2




 2 x
  u2
2 X
2  2

E x  
 exp
2
2
2  
 2 
Ex   X 
2

0


 

Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
6 of 15
ECE 3800
Cond. gaussian Dist. and Density Fundtions
1
0.9
0.8
0.7
0.6
pdf
PDF
c-pdf
C-PDF
0.5
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
1
The new variance is

 x X
Ex   x 
 exp
 2  2
2  
x  

 
X
2
2
2
   
0
Ex
2
x  

2

2
 
2
E x2  X  2 
2
2

4

  dx


  u2
 exp
uX 
2
2  
 2 
2
3

  du

  X   2
 
2

2
Var x   E x 2  E x    2  1  
 
see CondGaussianRandn.m
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
7 of 15
ECE 3800
Chapter 2 Examples
Problem 2-2.4: A random variable X has a probability distribution function of the form
Note: use sin not cos!
FX  x   0
 A  1  sin b  x 
1
  x  2
2  x  2
2 x
Determine the probability density function
f X x   0
  x  2
 A  b  cosb  x 
0
2  x  2
2 x
a) Find the values of A and b that make this a valid probability distribution function.
FX 2  A  1  sinb  2  1
FX  2  A  1  sinb  2  0
FX 2  FX  2  2  A  1
A 1
Therefore
2
FX  2  A  1  sin b  2  0
sinb  2  1
b2 
Therefore
2
and b  
FX  x   0

4
  x  2
1 
  
 1  sin  x    2  x  2
2 
 4 
1
2x
and
fX x   0
  x  2
 
 cos x  
8
 4
0


2  x  2
2 x
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
8 of 15
ECE 3800
Problem 2-2.4 Dist. and Density Fundtions
1
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0.8
0.7
0.6
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PDF
0.5
0.4
0.3
0.2
0.1
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
b) Find the probability that X>1
Pr  X  1  1  FX 1
Pr  X  1  1 - FX 1  1 
Pr  X  1  1 - FX 1 
1 
  
 1  sin   
2 
 4 
1 1
2 2
  1 1 1

 0.146
  sin     
2 2
4
4 2 2 2
c) Find the probability that X is negative
Pr  X  0  FX 0
Pr  X  0   F X 0  
1
1
 1  sin 0  
2
2
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
9 of 15
ECE 3800
d) Find the mean

X  E X  
x f
X
x   dx

X  EX  
2



 x  8  cos x  4   dx
2
1
 x  cosax  dx  a 2  cosax  a  sinax
x
2





1
 
  x
X  EX    
 cos x     sin  x  
8  2
 4 
 4 


4
 2
4
2
 
  x
X  E X    cos x     sin  x  

 4  2
 4 2
2
2
  2
 
  2
    2


 sin   2  
X  E X     cos 2     sin  2      cos  2   
4 2
4 
 4 2
 4  



X  E X   1  1  0
2 x
a2  x2  2



cos
ax

 sin ax 
a2
a3
32
Var  X   4  2

2
 x  cosax  dx 
see HW_2_2_4.m
Problem 2-2.4 Histogram
5000
4500
4000
Number of Samples
3500
3000
2500
2000
1500
1000
500
0
-2.5
-2
-1.5
-1
-0.5
0
0.5
Amplitude
1
1.5
2
2.5
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
10 of 15
ECE 3800
Problem 2-4.5: A random variable X has a probability density function of the form
f X x   a  x 2
0 x2
 ax
2 x3
a) Find the value a

1
 f X x  dx

2
3


2
1  a  x  dx  a  x  dx
0
2
x3
1 a
3
1 a
2
x2
a
2
0
3
2
23
03
32
22
 16  15 
8 9 4
a
a
a
 a      a 

3
3
2
2
 6 
3 2 2
a
6
31
a) Find the mean of the random variable X
X  E X  

x f
X
x   dx

X  E X  
2
xa x
3
2

 dx  x  a  x  dx
0
x4
X  EX   a 
4
X  EX   a 
2
2
0
x3
a
3
3
2
24
04
33
23
 16 27 8  6  48  76  124
2
a
a
a
 a  
   

4
4
3
3
3 3  31  12  62
 4
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
11 of 15
ECE 3800
Find the mean square or 2nd moment of the random variable X

X
2
   x
EX
2
2
 f X  x   dx

2
X
2
   x
EX
2
3
2

2
 a  x  dx  x 2  a  x  dx
0
X
 
X 2  E X 2  a
2
2
 
EX
2
x5
 a
5
2
0
x4
a
4
3
2
25
34
24
 32 81 16  6  128  325  1359
 4.383
a
a
 a      

5
4
4
4
4  31 
20
 310
 5
Find the variance or standard deviation
2  X2 X
2
 4.383  2 2  0.383
  0.620
c) Find the probability that 2<x≤3
Pr 2  X  3 
3
 f X x  dx
2
3

Pr 2  X  3  a  x  dx
2
x2


Pr 2  X  3  a 
2
3
2
32
22
6 9  4 15
 a
a
 

 0.484
2
2 31 2
31
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
12 of 15
ECE 3800
Find the probability that 2 - 0.620 < X ≤ 2 + 0.620
 
Pr     X      
 f X x  dx
 
2  0.62
2
Pr     X      
a x
2
 dx 
2  0.62
x3
Pr     X       a 
3
2
2  0.62
 a  x  dx
2
x2
a
2
2  0.62
2
23
1.383
2.62 2
22
a
a
a
Pr     X       a 
3
3
2
2
6  8 2.628 6.864 4 
  

 
31  3
3
2
2
6
  1.791  1.432 
31
 0.624
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
13 of 15
ECE 3800
Problem 2-5.5: A common method for detecting a signal in the presence of noise is to establish
a threshold level and compare the value of any observation with this threshold. If the
threshold is exceeded, it is decided that the signal is present. Sometimes, of course, noise alone
will exceed the threshold and this is known as a “false alarm” (also referred to as the “false
alarm rate” or FAR). Usually it is desired to have the probability of a false alarm very small.
At the same time, we should like any observation that does contain a signal plus the noise to
exceed the threshold with a large probability. Suppose we have Gaussian noise with zero mean
and variance of 1 Volt^2 and we set a threshold level of 5V.
a) Find the probability of a false alarm (FAR)
f X x  


 x X 2 
 ,for    x  
 exp


2
2  
 2 

1
f X x  
FX x  
  x2 

 exp


2
2


1
,for    x  
x
 u2 
  du   x 
exp 


2
2


u  
1


Threshold = 5, therefore X>5
FAR  1  FX 5  1  5  Q5  0.2867  10 6
b) If a signal having a value of 8 Volts is observed in the presence of this noise, find the
probability of detection (PD).
The signal plus noise is represented by a Gaussian distribution centered at 8V. The threshold for detection
remains at 5V (and above).
f S x  
FS  x  
   x  82 

 exp


2
2


1
,for    x  
x
 u  82 
  du  u  8

exp 


2
2


u  
1

Threshold = 5, therefore X>5 when mean(X)=8.
PD  PrX  5, X  8  1  FS 5  1  5  8  1   3  3  0.9987
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
14 of 15
ECE 3800
Expected Values of a Sinusoid with Random Phase
Let Y  a  cosw  t   where a, w, and t are constants, and  is a uniform random variable in
the interval (0,2π). The random variable Y results from sampling the amplitude of a sinusoid
with random phase  . Find the expected value of Y and expected value of the power of Y, Y2.
2
EY   Ea  cosw  t     a  cosw  t    
0
EY  
EY  
1
 d
2
a
2
 sinw  t    0
2
a
 sinw  t  2   sinw  t  0
2
EY  
a
 0  0
2
The average power is
   Ea
EY
2
2


 cosw  t     a 2  cosw  t    
2
2

 
EY2 
 
EY
2
2
1
 d
2

a
1 1

     cos2  w  t  2     d
2  2 2

2
a 2 
1 1
2 


   sin2  w  t  2    0 
2  2 0
2 2

 
EY2 
a2
2
2
 2 1 1  a
     0 
2 2 2  2
Note that these answers are in agreement with the time averages of sinusoids: the time average
(“dc” value) of the sinusoid is zero; the time-average power is a2/2.
Note: different results occur for a uniform distribution from (0, π)!
EY  
a

 sinw  t    0 

a

 sinw  t     sinw  t  0  
2a

 sinw  t 
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
15 of 15
ECE 3800