Tag: Bernoulli random variable

Binomial random variable using Matlab

Binomial random variable, a discrete random variable, models the number of successes in mutually independent Bernoulli trials, each with success probability . The term Bernoulli trial implies that each trial is a random experiment with exactly two possible outcomes: success and failure. It can be used to model the total number of bit errors in the received data sequence of length that was transmitted over a binary symmetric channel of bit-error probability .

Generating binomial random sequence in Matlab

Let X denotes the total number of successes in mutually independent Bernoulli trials. For ease of understanding, let’s denote success as ‘1’ and failure as ‘0’. Suppose if a particular outcome of the experiment contains ones and zeros (example outcome: 1011101), the probability mass function↗ of is given by

A binomial random variable can be simulated by generating independent Bernoulli trials and summing up the results.

function X = binomialRV(n,p,L)
%Generate Binomial random number sequence
%n - the number of independent Bernoulli trials
%p - probability of success yielded by each trial
%L - length of sequence to generate
X = zeros(1,L);
for i=1:L,
   X(i) = sum(bernoulliRV(n,p));
end
end

Following program demonstrates how to generate a sequence of binomially distributed random numbers, plot the estimated and theoretical probability mass functions for the chosen parameters (Figure 1).

n=30; p=1/6; %number of trails and success probability
X = binomialRV(n,p,10000);%generate 10000 bino rand numbers
X_pdf = pdf('Binomial',0:n,n,p); %theoretical probility density
histogram(X,'Normalization','pdf'); %plot histogram
hold on; plot(0:n,X_pdf,'r'); %plot computed theoreical PDF
PMF generated from binomial random variable for three different cases of n and p
Figure 1: PMF generated from binomial random variable for three different cases of n and p

PMF sums to unity

Let’s verify theoretically, the fact that the PMF of the binomial distribution sums to unity. Using the result of Binomial theorem↗,

Mean and variance

The mean number of success in a binomial distribution is

The variance is

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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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Bernoulli random variable

Bernoulli random variable is a discrete random variable with two outcomes – success and failure, with probabilities p and (1-p). It is a good model for binary data generators and also for modeling bit error patterns in the received binary data when a communication channel introduces random errors.

To generate a Bernoulli random variable X, in which the probability of success P(X=1)=p for some p ϵ (0,1), the discrete inverse transform method [1] can be applied on the continuous uniform random variable U(0,1) using the steps below.

  ● Generate uniform random number U in the interval (0,1)
  ● If U<p, set X=1, else set X=0

#bernoulliRV.m: Generating Bernoulli random number with success probability p
function X = bernoulliRV(L,p)
%Generate Bernoulli random number with success probability p
%L is the length of the sequence generated
U = rand(1,L); %continuous uniform random numbers in (0,1)
X = (U<p); end

Verifying law of large numbers

In probability theory, the law of large numbers is a theorem that involves repeating an experiment for a large number of times. According to this law, as the number of trials tend to become large, the average result obtained will be close to the expected value.

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

Let’s toss a coin with probability of heads . This experiment is repeated for a large number of times, say and the average result for each trial are calculated in a cumulative fashion.

#lawOfLargeNumbers.m: Law of large numbers illustrated using Bernoulli random variable
n=1000; %number of trials
p=0.7; %probability of success
X=bernoulliRV(n,p); %Bernoulli random variable
y_sum=sum(triu(repmat(X,[prod(size(X)) 1])')); %cumulative sum
avg = y_sum./(1:1:n); %average of results
plot(1:1:n,avg,'.'); hold on;
xlabel('Trial #'); ylabel('Probability of Heads');
plot(p*ones(1,n),'r'); legend('average','expected');

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

Figure 1: Illustrating law of large numbers using Bernoulli trials

The resulting plot (Figure 1) shows that as the number of trial increases, the average approaches the expected value 0.7.

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References

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

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
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Checkout Added to cart

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(PDF ebook)
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Digital Modulations using Matlab
(PDF ebook)
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Best books on Signal Processing

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