“Out of the 15 bank customers to whom the manager offered to connect autopayments, four agreed. Service activation is a binary feature that can be described by the Bernoulli distribution.”.

Let’s find the maximum likelihood estimate for the parameter p out of such a sample.

1) Likelihood function:

L(X_n, p) = ∏ p[X_i=1]*(1−p)[X_i=0] = p^4 * (1-p)^11

2) We find the maximum likelihood estimate for the parameter p.
We logarithm L(X_n, p) and get the following:

ln(p^4 * (1-p)^11) = 4*ln(p) + 11*ln(1-p)

3) Now we take its derivative and equate it to zero to find p.
[4ln(p) + 11ln(1-p)]` = 4 (ln(p))` + 11 (ln(1-p))` = 4/p + 11/(1-p) * (-1) = 0
Following: 4/p = 11/(1-p) => 4(1-p) = 11p => 15p = 4 => p = 4/15 =~ 0.26667.