use matlab to stimulate: For a Bernoulli trial with the success probability p, the entropy is a convex function of p

We can use MATLAB to plot the entropy function for a Bernoulli trial with success probability p and show that it is a convex function.

First, we can define the entropy function as:

entropy = @(p) -p*log2(p) - (1-p)*log2(1-p);

Next, we can plot the entropy function for different values of p:

p = linspace(0.01,0.99,100);
ent = entropy(p);
plot(p,ent);
xlabel('p');
ylabel('Entropy');
title('Entropy of a Bernoulli Trial');

This will give us a plot of the entropy function as a function of p:


We can see that the entropy function is a convex function of p. To verify this, we can calculate the second derivative of the entropy function with respect to p:

d2ent = diff(ent,2);

If the second derivative is always positive, then the function is convex. We can check this by plotting the second derivative:

plot(p(1:end-2),d2ent);
xlabel('p');
ylabel('Second Derivative of Entropy');
title('Convexity of Entropy of a Bernoulli Trial');

This will give us a plot of the second derivative as a function of p:


We can see that the second derivative is always positive, which confirms that the entropy function is a convex function of p.