Exponential random variable – simulation & application

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Introduction

An exponential random variable (RV) is a continuous random variable that has applications in modeling a Poisson process. Poisson processes find extensive applications in tele-traffic modeling and queuing theory. They are used to model random points in time or space, such as the times when call requests arriving at an exchange, the times when a shot noise occurs in the photon counting processing of an optical device, the times when file requests arrive at a serve etc.

Estimated PDF from an exponential random variable
Figure 1: Estimated PDF from an exponential random variable

Univariate random variables

This section focuses on some of the most frequently encountered univariate random variables in communication systems design. Basic installation of Matlab provides access to two fundamental random number generators: uniform random number generator (rand) and the standard normal random number generator (randn). They are fundamental in the sense that all other random variables like Bernoulli, Binomial, Chi, Chi-square, Rayleigh, Ricean, Nakagami-m, exponential etc.., can be generated by transforming them.

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● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

Exponential RV

An exponential random variable T takes value in the interval [0,\infty) and has the following continuous distribution function (CDF).

F_T(t) = 1-e^{-\lambda t} \quad\quad, t \in [0,\infty)

The rate parameter \lambda specifies the mean number of occurrences per unit time and t is the number of time units until the occurrence of next event that happens in the modeled process. The probability density function of the exponential rv is given by

f_T(t) = \lambda e^{-\lambda t}, \quad \quad t \in [0,\infty)

By applying the inverse transform method [1], an uniform random variable U(0,1) can be transformed into an exponential random variable. This method is coded in the Matlab function that is shown at the end of this article. Using the function, a sequence of exponentially distributed random numbers can be generated, whose estimated pdf is plotted against the theoretical pdf as shown in the Figure 1.

F^{-1}(u) = \frac{-ln \left( 1- u\right)}{\lambda}, \quad\quad u \in [0,1)

Application to Poisson process

Poisson process is a continuous-time discrete state process that is widely used to model independent events occurring in time or space. It is widely applied to model a counting process in which the events occur at independently random times but appear to happen at certain rate. In practice, Poisson process has been used to model counting processes like

  • photons landing on a photo-diode
  • arrivals of phone calls at a telephone exchange
  • request for file downloads at a web server
  • location of users in a wireless network

Poisson process is closely related to a number of vital random variables (RV) including the uniform RV, binomial RV, the exponential RV and the Poisson RV. For example, the inter-arrival times (duration between the subsequent arrivals of events) in a Poisson process are independent exponential random variables.

For example, the inter-arrival times (duration between the subsequent arrivals of events) in a Poisson process are independent exponential random variables

Refer the book Wireless Communication Systems in Matlab for full Matlab code

function T = expRV(lambda,L)
%Generate random number sequence that is exponentially distributed
%lambda - rate parameter, L - length of the sequence generated
U = rand(1,L); %continuous uniform random numbers in (0,1)
T = -1/lambda*(log(1-U));
end

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References

● L. Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, New York, 1986.↗

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Topics in this chapter

Random Variables - Simulating Probabilistic Systems
● Introduction
● Plotting the estimated PDF
● Univariate random variables
 □ Uniform random variable
 □ Bernoulli random variable
 □ Binomial random variable
 □ Exponential random variable
 □ Poisson process
 □ Gaussian random variable
 □ Chi-squared random variable
 □ Non-central Chi-Squared random variable
 □ Chi distributed random variable
 □ Rayleigh random variable
 □ Ricean random variable
 □ Nakagami-m distributed random variable
● Central limit theorem - a demonstration
● Generating correlated random variables
 □ Generating two sequences of correlated random variables
 □ Generating multiple sequences of correlated random variables using Cholesky decomposition
● Generating correlated Gaussian sequences
 □ Spectral factorization method
 □ Auto-Regressive (AR) model

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