The binomial Formulla ~ University Notes

In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of

independent yes/no experiments, each of which yields success with probability

. The binomial distribution is the basis for the popular binomial test of statistical significance.

Binomial Probability Distribution

This is a graphic representation of a binomial probability distribution.

The binomial distribution is frequently used to model the number of successes in a sample of size

drawn with replacement from a population of size

. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for

much larger than

, the binomial distribution is a good approximation, and widely used.

In general, if the random variable

follows the binomial distribution with parameters

and

, we write

. The probability of getting exactly

successes in

trials is given by the Probability Mass Function:

For

where:

Is the binomial coefficient (hence the name of the distribution) "n choose k," also denoted

. The formula can be understood as follows: We want

successes (

) and

failures (

); however, the

successes can occur anywhere among the

trials, and there are

different ways of distributing

successes in a sequence of

trials.

One straightforward way to simulate a binomial random variable

is to compute the sum of

independent 0−1 random variables, each of which takes on the value 1 with probability

. This method requires

calls to a random number generator to obtain one value of the random variable. When

is relatively large (say at least 30), the Central Limit Theorem implies that the binomial distribution is well-approximated by the corresponding normal density function with parameters

and

Figures from the Example

Table 1

These are the four possible outcomes from flipping a coin twice.

Table 2

These are the probabilities of the 2 coin flips.