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The R Book by Michael J. Crawley

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Discrete probability distributions

The Bernoulli distribution

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:

images

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

images

and expectation of x2 is

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so the variance of the Bernoulli is

images

The binomial distribution

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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