## The binomial distribution discrete trials and discrete outcomes

We use the binomial distribution to describe a situation in which the experiment or observation is discrete (for example, the number of Steller sea lions Ewmaiopzas jwbatws who produce offspring, with one pup per mother per year) and the outcome is discrete (for example, the number of offspring produced). The key variable underlying a single trial is the probability p of a successful outcome. A single trial is called a Bernoulli trial, named after the famous probabilist Daniel Bernoulli (see Connections in both Chapter 2 and here). If we let X, denote the outcome of the ith trial, with a 1 indicating a success and a 0 indicating a failure then we write

Virtually all computer operating systems now provide random numbers that are uniformly distributed between 0 and 1; for a uniform random number between 0 and 1, the probability density is fz) = 1 if 0 < z < 1 and is 0 otherwise. To simulate the single Bernoulli trial, we specify p allow the computer to draw a uniform random number U and if

1 with probability p 0 with probability 1 — p

U <p we consider the trial a success; otherwise we consider it to be a failure.

The binomial distribution arises when we have N Bernoulli trials. The number of successes in the N trials is

This equation also tells us a good way to simulate a binomial distribution, as the sum of N Bernoulli trials.

The number of successes in N trials can range from K = 0 to K=N, so we are interested in the probability that K = k. This probability is given by the binomial distribution

In this equation is called the binomial coefficient and represents the number of different ways that we can get k successes in N trials. It is read "Nchoose k'' and is given by = N!/k!(N — k)!, where N! is the factorial function.

We can explore the binomial distribution through analytical and numerical means. We begin with the analytical approach. First, let us note that when k = 0, Eq. (3.22) simplifies since the binomial coefficient is 1 and p0 = 1:

This is also the beginning of a way to calculate the terms of the binomial distribution, which we can now write out in a slightly different form as

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