Mathcad For In Class Examples In A Random Processes Course

2006-812: MATHCAD FOR IN-CLASS EXAMPLES IN A RANDOM PROCESSESCOURSE

James Reising, University of EvansvilleJAMES A. REISING is an Associate Professor of Electrical Engineering at the University ofEvansville, Evansville, Indiana, where he has taught since 1980. Prior to that time he wasemployed by Eagle-Picher Industries at the Miami Research Laboratories and the Electro-OpticMaterials Department. He is a senior member of IEEE.

© American Society for Engineering Education, 2006

1.913.1

Mathcad™ for In-class Examples in a Random Processes Course

Abstract

Some textbooks1,2 used for courses in Probability and Random Processes use MATLAB™ for

computer demonstrations and exercises, while others3,4,5

do not emphasize specific computer

exercises or examples, and at least one6 uses C programs. I find it convenient to use Mathcad for

lecture examples. The appearance of calculations onscreen is practically identical to equations

appearing in the text, and graphs of illustrative results appear on the same page as the equations.

Calculations are all "live", causing graphs to be updated when the values of one or more

parameters are changed. This rapid response provides faster visual representation of important

concepts.

This paper gives examples of Mathcad worksheets for use during lecture in a course discussing

probability and random processes. The ability to easily and quickly change parameters used in

the equations with automatic updating of graphical display of results helps illustrate the concepts

involved.

Background

EE 413 Random Signals and Noise has been offered three times at the University of Evansville.

By the second offering it had become obvious that students found many of the concepts in

probability and random processes difficult to grasp, and the need for additional in-class examples

was apparent. I used Mathcad examples in class for the first time in the fall semester 2005.

The textbooks1,2 used (in different semesters) both use MATLAB in sample problems and

homework. MATLAB is used as well in a number of other courses in the EE curriculum and is

available on the department network, so students need not purchase the program. I have found it

useful to use Mathcad during lecture for illustrating the role of various parameters in

probabilistic examples for the reasons cited in the abstract section above. I use Mathcad only for

lecture examples, so students need not use or purchase the software.

Probability Density and Distribution Functions

The probability density function of a random variable X, )(xf X , and the cumulative probability

distribution function of X, )(xFX , are key concepts in the study of random variables. Mathcad

includes built-in functions for common distributions of both discrete and continuous random

variables, including the discrete binomial and Poisson distributions, and the continuous

exponential, Cauchy, and Gaussian (normal) distributions. All these functions include one or

more parameters characterizing a particular instance of the distribution. Simple plots of the

density and distributions functions can be used in class to examine the effects of varying the

values of the parameters.

Page 11.913.2

Figure 1 Figure 2

Figures 1 and 2 above illustrate a Mathcad worksheet using two different values of the parameter

p for the binomial distribution. Since the worksheet calculations are “live”, all the plots are

updated as soon as the parameter value is changed at the top of the page. By observing the plots

for several different values of p, one not only sees how the distribution functions change, but

may also notice that the peak of the density function occurs at or near a value that is the product

of n and p. The theoretical result that the mean value of a random variable having a binomial

distribution is np then seems quite reasonable. The Mathcad functions dbinom and pbinom

return values of the probability density and cumulative distribution, respectively. The third plot

shows that summing the probability density function gives the same result for the cumulative

distribution function as the built-in function.

Figures 3 and 4 below illustrate similar results for two values of the parameter α in the Poisson

distribution.

Figure 3 Figure 4

Binomial Distribution

n 20:= p .7:= k 0 n..:=

1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 210

0.1

0.2

dbinom k n, p,( )

k

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.5

1

pbinom k n, p,( )

k

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.5

1

0

k

i

dbinom i n, p,( )∑=

k

Binomial Distribution

n 20:= p .2:= k 0 n..:=

1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 210

0.1

0.2

dbinom k n, p,( )

k

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.5

1

pbinom k n, p,( )

k

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.5

1

0

k

i

dbinom i n, p,( )∑=

k

Poission Random Variables

j 1− 25..:=

λ .1:= T 20:= α λ T⋅:= α 2=

1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250

0.2

dpois j α,( )

j

1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250

0.5

1

ppois j α,( )

j

Poission Random Variables

j 1− 25..:=

λ .8:= T 20:= α λ T⋅:= α 16=

1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250

0.2

dpois j α,( )

j

1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250

0.5

1

ppois j α,( )

j Page 11.913.3

A discrete probability distribution for which Mathcad has no built-in functions is the Pascal

distribution given in the text by Yates and Goodman2 , where kxk

X ppk

xxf −−

−

−= )1(

1

1)( for

10 << p , k is an integer such that 1≥k , and x is an integer. This may be expressed in a form

that relates it to the binomial distribution as kxk

X ppk

x

x

kxf −−

= )1()( , which is just

x

ktimes the

binomial density function. The Pascal density function should be zero for values of x less than k.

Figure 5 Figure 6

Figures 5 and 6 above show a Mathcad worksheet for two different combinations of values of p

and k.

Probability density and cumulative distribution functions for continuous random variables may

be displayed and examined in much the same way. For continuous random variables, however,

the probability density function gives the probability that the random variable has a value in an

infinitesimal interval about a particular value. To display the probability that a random variable

assumes a value within a finite range of values, a histogram is useful. While the probability

density function for a continuous random variable may assume a value greater than one at some

point, the probability that the variable has a value within a certain range must always be less than

or equal to one. The Mathcad worksheet in Figures 7 and 8 below illustrates this feature of

continuous distributions as well as the generation of a sequence of random numbers with a

particular probability distribution.

Pascal Random Variable as in Yates and Goodman

p .4:= k 7:= x 0 40..:=

0 10 20 30 400

0.05

0.1

k

xdbinom k x, p,( )⋅

x

Pascal Random Variable as in Yates and Goodman

p .5:= k 4:= x 0 40..:=

0 10 20 30 400

0.1

0.2

k

xdbinom k x, p,( )⋅

x

Page 11.913.4

Figure 7 Figure 8

This worksheet uses the built-in function rnorm to generate a vector of random numbers having a

normal distribution. It also uses the histogram function provided in Mathcad. The histogram

function produces a two-column matrix where the first column (h<0>

in Mathcad notation)

contains the bin centers for the histogram, and the second column (h<1>

) contains the frequencies

with which the data falls into the bins. The plots may be used to examine several key points.

Observing the change in the shape of the distribution due to variations in the mean and standard

deviation, µ and σ, respectively, provides some feel for what these parameters mean.

Contrasting the plot of the density function and the solid line representing theoretical

probabilities on the histogram gives some insight into the difference between the probability

density and the probability that a continuous random variable assumes a value within a certain

range. Changing the number of data points and comparing the actual mean and histogram of

random numbers to theoretical values illustrates the relationship between theory and sample size.

Figures 9 and 10 below show the same worksheet with changes in the distribution parameters.

The gaussian, or normal, probability distribution

n 50:= the number of random numbers to generate

L 3:= the absolute value of the maximum argument

µ 1:= the mean value σ .25:= the standard deviation

note the pdf may be greater than 1 dnorm µ µ, σ,( ) 1.596=

i 0 20 L⋅ 1+..:= intvlsi

i

10L−:=

v rnorm n µ, σ,( ):= the vector of random numbers mean v( ) 1.044=

h histogram intvls v,( ):= q pnorm h0⟨ ⟩

.05+ µ, σ,( ) pnorm h0⟨ ⟩

.05− µ, σ,( )−:=

First look at the density function

3 2 1 0 1 2 30

1

2

dnorm h0⟨ ⟩ µ, σ,( )

h0⟨ ⟩

Then look at histogram and randomly generated numbers

3 2 1 0 1 2 30

0.06

0.12

0.18

0.24

0.3

h1⟨ ⟩

n

q

h0⟨ ⟩

h0⟨ ⟩,




µ 1:= the mean value σ .25:= the standard deviation


i 0 20 L⋅ 1+..:= intvlsi

i

10L−:=



.05+ µ, σ,( ) pnorm h0⟨ ⟩

.05− µ, σ,( )−:=


3 2 1 0 1 2 30

1

2


h0⟨ ⟩


3 2 1 0 1 2 30

0.06

0.12

0.18

0.24

0.3

h1⟨ ⟩

n

q

h0⟨ ⟩

h0⟨ ⟩,

Page 11.913.5

Figure 9 Figure 10

Not all distribution functions of interest are Mathcad built-in functions. The Rayleigh density

function1 is one example:

b

x

xeb

xf X

22)(

−= bx

exFX

2

1)(−−= for x>0, b>0

Making use of the fact that1 solving the equation xyFY =)( for y as a function of x, where

)(yFY is the cumulative distribution function for the random variable Y, allows one to generate

random numbers y from uniformly distributed numbers x, 0 < x < 1. The Mathcad worksheet

shown as Figure 11 illustrates this example and compares the results to theory.




µ .5:= the mean value σ 1:= the standard deviation


i 0 20 L⋅ 1+..:= intvlsi

i

10L−:=



.05+ µ, σ,( ) pnorm h0⟨ ⟩

.05− µ, σ,( )−:=


3 2 1 0 1 2 30

0.2

0.4


h0⟨ ⟩


3 2 1 0 1 2 31 .10 4

0.02

0.04

0.06

0.08

0.1

h1⟨ ⟩

n

q

h0⟨ ⟩

h0⟨ ⟩,




µ .5:= the mean value σ 1:= the standard deviation


i 0 20 L⋅ 1+..:= intvlsi

i

10L−:=



.05+ µ, σ,( ) pnorm h0⟨ ⟩

.05− µ, σ,( )−:=


3 2 1 0 1 2 30

0.2

0.4


h0⟨ ⟩


3 2 1 0 1 2 30

0.02

0.04

0.06

0.08

0.1

h1⟨ ⟩

n

q

h0⟨ ⟩

h0⟨ ⟩,

Page 11.913.6

Generating random numbers with Rayleigh distribution

b 2= the Rayleigh parameter n 100000= number of numbers to generate

FX x( ) 1 e

x2−

b−= is the CDF

x runif n 0, 1,( )= array of n uniformly distributed numbers between 0 and 1

y b− ln 1 x−( )⋅→

= array of n Rayleigh distributed numbers

L 5= maximum argument for plot i 0 10 L⋅..= intvlsi

i

10.05+=

h histogram intvls y,( )= mean y( ) 1.254=

Theoretical results are as below:

µπ b⋅

4= µ 1.253=

q e

h0⟨ ⟩

.05−( )2−

be

h0⟨ ⟩

.05+( )2−

b−=

0 1 2 3 4 50

0.05

h1⟨ ⟩

n

q

h0⟨ ⟩

Figure 11

As in the earlier examples, changes made in the Rayleigh parameter b and/or the length n of the

data vector are reflected in the corresponding plot.

Pairs of Random Variables

The joint probability function is used in computing probabilities and expected values of

functions. Calculations involve the evaluation of double integrals. Figure 12 below shows a

very simple example of a joint probability function and multiple integral evaluation.

Page 11.913.7

Figure 12

A commonly discussed1,2 joint probability density function is that of bivariate, or jointly

Gaussian, random variables. Examining probability density function plots with different

correlation coefficient values provides some indication its role. Figure 13 shows a Mathcad

worksheet with nearly uncorrelated variables, while Figure 14 shows the same worksheet with

strongly correlated variables. In each of the two cases, the means and standard deviations are the

same.

Joint PDF example of multiple integral evaluation

C

0

1

y

0

1

x1 x y⋅−( )⌠⌡

d⌠⌡

d:= C3

4= required normalization

fXY x y,( )1

C1 x y⋅−( )⋅:= for 0 x≤ 1≤ and 0 y≤ 1≤ , 0 otherwise

check:

0

1

y

0

1

xfXY x y,( )⌠⌡

d⌠⌡

d 1=

0

1

x

0

1 x−

yfXY x y,( )⌠⌡

d⌠⌡

d11

18= is P(x+y < 1)

0

1

x

1 x−

1

yfXY x y,( )⌠⌡

d⌠⌡

d7

18= is P(x+y > 1)

0

1

x

x

1

yfXY x y,( )⌠⌡

d⌠⌡

d1

2= is P(y > x)

0

1

x

0

x

yfXY x y,( )⌠⌡

d⌠⌡

d1

2= is P(y < x)

0

1

y

0

1

xx 1 x y⋅−( )⋅⌠⌡

d⌠⌡

d1

3= is the mean value of x

0

1

y

0

1

xx y2

⋅ 1 x y⋅−( )⋅⌠⌡

d⌠⌡

d1

12= is the mean value of xy2

Page 11.913.8

Figure 13 Figure 14

White Noise Simulation and Autocorrelation Sequence

A final example uses a sequence of random numbers with Gaussian distribution as simulated

white noise and computes the autocorrelation sequence. Theoretically an infinite sequence of

such numbers would have an autocorrelation sequence with a single non-zero value at zero offset

equal to the variance of the Gaussian distribution. This value indicates the power per unit

frequency in the white noise. A finite sequence used to simulate a white noise sequence will

have an autocorrelation sequence exhibiting the effects of the finite (rectangular-windowed)

sequence used. Figures 15 and 16 on the following pages show the Mathcad worksheet. Figure

15 uses a sequence of 10,000 random numbers as simulated noise. The effect of the finite

sequence length is barely noticeable at the vertical scale of the autocorrelation plot. Figure 16,

on the other hand, uses a sequence of only 100 random numbers, and the effects of the finite

sequence length are clearly visible on the corresponding autocorrelation plot. In-class

experimentation with different sequence lengths provides an indication of the relationship

between sequence length and its effect on the autocorrelation sequence.

Jointly Gaussian Random Variable PDF

mx 0:= my 1:= mean values

σx 1:= σy 1.5:= standard deviations

ρ .1:= correlation coefficient

i 50− 49−, 50..:= j 50− 49−, 50..:=

fXYi j,

1

2 π⋅ σx⋅ σy⋅ 1 ρ2

−⋅

e

1−

2 1 ρ2−( )⋅

i

10mx−

2

σx2

2 ρ⋅i

10mx−

⋅j

10my−

⋅

σx σy⋅−

j

10my−

2

σy2

+

⋅

⋅:=

fXY

Jointly Gaussian Random Variable PDF

mx 0:= my 1:= mean values

σx 1:= σy 1.5:= standard deviations

ρ .9:= correlation coefficient

i 50− 49−, 50..:= j 50− 49−, 50..:=

fXYi j,

1

2 π⋅ σx⋅ σy⋅ 1 ρ2

−⋅

e

1−

2 1 ρ2−( )⋅

i

10mx−

2

σx2

2 ρ⋅i

10mx−

⋅j

10my−

⋅

σx σy⋅−

j

10my−

2

σy2

+

⋅

⋅:=

fXY

Page 11.913.9

White Noise Simulation and its Autocorrelation Sequence

First generate a sequence of zero-mean Gaussian random variables

n 10000:= is the length of the sequence x rnorm n 0, 5,( ):= length x( ) 1 104

×=

Plot the sequence as bars m 0 length x( )..:=

0 2000 4000 6000 8000 1 .10420

0

20

xm

m

Compute and plot the autocorrelation sequence. Note the zero-offset value is the first element

in the array, next are the positive offsets, followed by the negative offsets. The sequence must

be rearranged fo a plot to show zero-offset value in the center, as is customary.

rcorrel x x,( )

length x( ):= i 0 length r( ) 1−..:= r

024.83= length r( ) 2 10

4×=

0 5000 1 .104 1.5 .10420

0

20

40

ri

iFinally plot the autocorrelation sequence in customary form after rearranging it.

foldlength r( ) 1−

2:= j 0 fold 1−..:= k fold length r( ) 1−..:=

dk

rk fold−:= d

jrj fold+ 1+:=

0 5000 1 .104 1.5 .10420

0

20

40

di

i

Figure 15

Page 11.913.10

White Noise Simulation and its Autocorrelation Sequence

First generate a sequence of zero-mean Gaussian random variables

n 100:= is the length of the sequence x rnorm n 0, 5,( ):= length x( ) 100=

Plot the sequence as bars m 0 length x( )..:=

0 20 40 60 80 10020

0

20

xm

m

Compute and plot the autocorrelation sequence. Note the zero-offset value is the first element

in the array, next are the positive offsets, followed by the negative offsets. The sequence must

be rearranged fo a plot to show zero-offset value in the center, as is customary.

rcorrel x x,( )

length x( ):= i 0 length r( ) 1−..:= r

030.769= length r( ) 199=

0 50 100 15050

0

50

ri

iFinally plot the autocorrelation sequence in customary form after rearranging it.

foldlength r( ) 1−

2:= j 0 fold 1−..:= k fold length r( ) 1−..:=

dk

rk fold−:= d

jrj fold+ 1+:=

0 50 100 15020

0

20

40

di

i

Figure 16

.913.11

Conclusion

The examples presented here illustrate several ways in which important concepts in probability

and random processes can be demonstrated in the classroom. The effects of changes in model

parameters are easy to investigate due to the “live” calculations of Mathcad worksheets. The

graphical user interface with its standard mathematical symbols provides an additional

advantage.

Students unanimously stated in informal evaluations that they found the additional Mathcad

examples helpful and thought their use should be continued.

Mathcad 13 examples used in this paper are available at http://csserver.evansville.edu/~reising .

Bibliography

1. Peebles, Peyton Z., Jr. Probability, Random Variables and Random Signal Principles 4th Edition.

McGraw-Hill, Inc., 2001.

2. Yates, Roy, and David Goodman. Probability and Stochastic Processes A Friendly Introduction for

Electrical and Computer Engineers 2nd Edition. John Wiley & Sons, Inc., 2005.

3. Papoulis, Athanasios, and S. Unnikrishna Pillai. Probability, Random Variables and Stochastic Processes

Fourth Edition. Boston, MA: McGraw-Hill, 2002.

4. Stark, Henry, and John W. Woods. Probability and Random Processes with Applications to Signal

Processing 3rd Edition. Upper Saddle River, NJ: Prentice-Hall, 2002.

5. Viniotis, Yannis. Probability and Random Processes for Electrical Engineers. Boston, MA:

WCB/McGraw-Hill, 1998.

6. Leon-Garcia, Alberto. Probability and Random Processes for Electrical Engineering 2nd Edition. Reading,

MA: Addison-Wesley Longman, 1994.

Page 11.913.12

Mathcad For In Class Examples In A Random Processes Course

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