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 x2 minus the square of the expectation of x : σ2 = E(X2) − [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 x2 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 ...