Bernoulli random variable

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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.

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.

References

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

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