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 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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