A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution. When you flip a coin, there are two possible outcomes: heads and tails. Each outcome has a fixed probability, the same from trial to trial. In the case of coins, heads and tails each have the same probability of 1/2. More generally, there are situations in which the coin is biased, so that heads and tails have different probabilities. In the present section, we consider probability distributions for which there are just two possible outcomes with fixed probabilities summing to one. These distributions are called binomial distributions.

Example: A typist makes on average 2 mistakes per page. What is the probability of a particular page having no errors on it?

We have an average rate here: lambda = 2 errors per page.

We don't have an exact probability (e.g. something like "there is a Probability of 1/2 that a page contains errors").

Hence, Poisson distribution.

(Lambda t) = (2 errors per page * 1 page) = 2.

Hence P0 = 2^0/0! * Exp (-2) = 0.135.

POISSON DISTRIBUTION

The Poisson distribution can be used to calculate the probabilities of various numbers of "successes" based on the mean number of successes. In order to apply the Poisson distribution, the various events must be independent. Keep in mind that the term "success" does not really mean success in…