DEMO PROGRAMS FOR NORMAL AND UNIFORM

RANDOM VARIABLES TRANSFORMATION

Gordana Jovanovic Dolecek and Fred Harris

Department of Electronics, Institute INAOE, Puebla, Mexico

Department of Electrical Eng., SDSU San Diego, U.S.A.

Keywords: Demo program, MATLAB, Linear transform, Nonlinear transform, Normal variables, Uniform variables.

Abstract: This paper presents MATLAB-based demo programs for transformation of normal and uniform random

variables. Linear as well as nonlinear transforms are considered. It is demonstrated how the transform

changes the corresponding probability density function. It is also shown how to use the transformation of

given random variable to generate a new desired random variable. The programs can be used as a

complement to theoretical classes or alone as a self-study tool.

1 INTRODUCTION

Computer-aided learning has become an important

educational research activity in various engineering

disciplines and there has been a growing interest in

the development of educational software in all areas

of study (Fernandez and Sanchez, 2004). As a result

many computer packages have been developed to

assist learning (Yann and Teng, 2003; Li and Lie,

2004).

It is known that random variable (r.v.) is generally

considered as one of the most abstract and

conceptually difficult areas in the engineering

education and teaching of random variables is one of

the subjects that requires more time for its

understanding. Our experience shows that the use of

demo programs gives students the visual and

intuitive representation of the random variables

which had traditionally been stated in terms of

abstract mathematical description (Jovanovic, 1997;

Jovanovic and Champac, 2000). Such software tools

must have characteristics like repeatability, capacity

to motivate students, interactivity, versatility, easy to

use etc.

Presented in this paper are demo programs to

teach the transform of uniform and normal random

variables. The programs can be used as a

complement to theoretical classes or alone as a self-

study tool. The programs are interactive, i.e., the

user chooses the parameters of the input variables

and the corresponding transform.

Each program presents input and output variables

and the desired transform as well as the estimated

probability densities. The exact densities are also

presented.

The rest of the paper is as follows. Next section is

devoted to the linear and nonlinear transformations

of uniform random variables, while Section 3

demonstrates the linear and nonlinear transforms of

normal random variables.

2 UNIFORM VARIABLE

2.1 Demo for Linear Transform

In this program the linear transform (LT) Y=aX+b of

the uniform random variable X is discussed. The

user chooses the range (R

, R

) of the uniform r.v.

and the parameters of the linear transform a and b.

The program shows the output variable and the

estimation of the input and output probability

densities (PDS), thus demonstrating that the linear

transform of the uniform r.v. results in an uniform

r.v. As an illustration Fig.1 shows the result for R

-1, R

= 4, a = 2, and b = 3. The estimation of the

corresponding PDFs is given in Fig.2.

Figure 3 presents the corresponding uniform

PDFs.

It is also demonstrated how this result is useful to

generate the desired uniform r.v. from the given

uniform r.v. For example, in MATLAB the file rand

363

Dolecek G. and Harris F. (2009).

DEMO PROGRAMS FOR NORMAL AND UNIFORM RANDOM VARIABLES TRANSFORMATION.

In Proceedings of the First International Conference on Computer Supported Education, pages 363-366

DOI: 10.5220/0001835003630366

 SciTePress

generates the uniform r.v. in the interval (0, 1). To

obtain uniform r.v. in the interval (R

, R

), we have

to apply the linear transform with a=R

-R

, where

a=R

-R

; b=R

, to the uniform random variable

generated by rand. Figure 4 illustrates the generation

of the uniform r.v. in the interval (-3, 5).

Figure 1: Linear transform of uniform r.v.

Figure 2: Estimated Input and Output Densities.

Figure 3: Input and Output Densities.

2.2 Demo for Nonlinear Transform

The Demo program for the absolute value of

uniform r.v. illustrates that the transform does not

change the type of the uniform r.v., as shown in

example in Fig.5, where the input r.v. is uniform in

the interval (-5, 5). The estimation of the input and

output PDFs are given in Fig.6.

The following demo shows the transformations

of the uniform r.v. X in the range (0, 1).

The demo illustrated in Figs 7 and 8 shows how

to generate an exponential random variable by

applying transform

log(1 )YX

=− −

(1)

to a uniform variable X.

Figure 9 illustrates the demo for the

transformation of the uniform random variable X,

2lnYX

=− .

(2)

Figure 4: Generation of the desired uniform r.v.

Figure 5: Absolute value of uniform r.v.

The resulting random variable is Rayleigh as

shown in Fig.9.

CSEDU 2009 - International Conference on Computer Supported Education

364

3 NORMAL VARIABLE

3.1 Demo for Linear Transform

This program demonstrates that the linear transform

of the normal r.v. gives again the normal r.v. User

chooses the mean value and variance of the normal

variable, as well as the parameters of the linear

transform a and b. Figure 10 illustrates the demo for

the mean value 2, variance 4, and a=3, b=4. The

figure shows the estimation of the output PDF

demonstrating that the output random variable is

also the normal variable. The same is confirmed in

Fig.11.

MATLAB file randn generates the normal r.v.

with the zero mean value and the variance 1. The

normal random variable with mean m and the

variance

is obtained by applying the following

transform

YXm

=+.

(3)

Figure 6: Estimation of the corresponding densities.

Figure 7: The generation of exponentional r.v.

Figure 8: Estimation of the corresponding densities.

Figure 9: Estimation of the Rayleigh density.

Figure 10: Estimation of the output density.

3.2 Demo for Nonlinear Transform

Figure 12 illustrates the absolute value of the normal

r.v. with zero mean and the variance 4. The resulting

random variable is the one-sided normal variable.

The exponential transform

(4)

of the normal variable is illustrated in Figs.13 and

14.

DEMO PROGRAMS FOR NORMAL AND UNIFORM RANDOM VARIABLES TRANSFORMATION

365

4 CONCLUSIONS

This paper describes the educational software for

transformations of the uniform and normal random

variables. The linear as well as nonlinear transforms

are considered. It is also described how to use the

transformations to generate different random

variables.

Figure 11: Input and output densities.

Figure 12: Absolute value of normal r.v.

Figure 13: Exponential transform.

Figure 14: Estimation of the densities.

The programs are written in MATLAB, although

any previous knowledge of MATLAB is not

required. The programs can be used as a

complement to theoretical classes or alone as a self-

study tool for teaching basic course on random

signals.

ACKNOWLEDGEMENTS

This work is supported by CONACYT grant No.

91013.

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