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Challenge Data Mining

Finding maximum likelihood estimate for the Bernoulli distribution parameter

“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(Xn, p) = ∏ p[Xi=1]*(1−p)[Xi=0] = p^4 * (1-p)^11

2) We find the maximum likelihood estimate for the parameter p.
We logarithm L(Xn, 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.

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