This is the distribution underlying tests with a binary response variable. The response takes one of only two values: it is 1 with probability *p* (a ‘success’) and is 0 with probability 1 – *p* (a ‘failure’). The density function is given by:

The statistician's definition of **variance** is the expectation of *x*^{2} minus the square of the expectation of *x* : σ^{2} = E(*X*^{2}) − [E(*X*)]^{2}. We can see how this works with a simple distribution like the Bernoulli. There are just two outcomes in *f(x)*: a success, where *x* =1 with probability *p* and a failure, where *x* = 0 with probability 1 – *p*. Thus, the expectation of *x* is

and expectation of *x*^{2} is

so the variance of the Bernoulli is

This is a one-parameter distribution in which *p* describes the probability of success in a binary trial. The probability of *x* successes out of *n* attempts is given by multiplying together the probability of obtaining one specific realization and the number of ways of getting that realization.

We need a way of generalizing the number of ways of getting ...

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