“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**.