Bernoulli Process and Coin Tosses

The Distribution

Jacob Bernoulli

The Bernoulli distribution is an example of a discrete distribution. The paradigm is to observe the result X of a coin toss, where X is 1 if the result is heads or 0 otherwise. Then, by definition X has a Bernoulli distribution. This distribution is a good model for any experiment where we are interested to know whether a particular event happened or not.

Going back to our coin, where X has two possible values X∈{0,1}, the probabilities for each value are defined by:

P(X =0 0) = 1 − p, P(X = 1) = p

In the case of an unbiased coin, p = 0.5 and both possible events have the same chance of occurrence.

Moments of the Bernoulli Distribution

Variance for the Bernoulli distribution 

The first moment or expected value of the distribution is:

<X> =  0⋅(1 − p) + 1⋅p = p

The second moment or expected value of the square of X is:

<X²> = 0²⋅(1 − p) + 1²⋅p = p

Therefore, the variance for a variable distributed according to the Bernoulli distribution is:

σ² = <X>² − <X²> = p - p² = p⋅(1 − p)

Considering the variance as a function of p:

f(p) = p⋅(1 − p)

It is easy to see that the value of p making the variance a maximum (p_max) is:

f'(p_max) = 0 = -2⋅p_max +1  ∴  p_max = 0.5

Where f'(p_max) stands for the derivative of f respect to p evaluated at p_max (the maximum).