Bernoulli random variable

by

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.

[table id = 36/]

Let’s toss a coin with probability of heads $latex p=0.7$. This experiment is repeated for a large number of times, say $latex n=1000$ 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

Illustrating law of large numbers using Bernoulli trials
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.

Rate this article: [ratings]

References

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

Books by the author

[table id=23 /]

Topics in this chapter

[table id=33 /]

Leave a Comment