Bernuoli distribution details (including detailed derivation steps for Digital features of the distribution)

Bernouli Distribution(Chinese Translation: bernuoli distribution)

This distributed study is a special experiment in which only two results are either successful or failed, and each experiment is independent and each experiment has a fixed probability of success p.

Probability formula can be expressed as, X can only be 0 or 1, that is, either success or failure

Based on the expected nature of Mathematics

 

Since X only has two values, the mathematical expectation for this distribution is

The variance can be calculated by the variance formula.

Variance formula:

The distribution is obvious, so we can get,

Therefore, variance

 

Finally, we will deduce the maximum likelihood estimation of the distribution.

It is defined in this way. If we have performed n experiments and the result set is, we want to find one that makes the set the most likely, because each experiment is independent, you can use the multiplication principle.

Since 0 <p <1, to maximize the formula above, it is equivalent to maximizing the formula subvalue with the ln base, we can convert the concatenation into the form of addition with the ln base, the formula is as follows:

We can see from the maximum likelihood equation that to select P to maximize this formula, we only need to make this formula evaluate P to 0, that is

,

After simplification, we finally get

This is the maximum likelihood estimation of the bernuoli distribution.