We have seen a few discrete probability distributions by now. Today we summarise them and find the relationships and the differences. The following are considered here:
Bernoulli distribution
The Bernoulli distribution is the distribution of the number of successes on a single Bernoulli trial. In a Bernoulli trial, you either get a success (1) or a failure (0). Therefore, a Bernoulli random variable can take either zero or one. E.g., if a coin is tossed once, what is the probability that it comes up heads?
Binomial distribution
When you carry out multiple Bernoulli trials, we get into a Binomial distribution. E.g., if I toss a coin ten times, what is the probability of getting exactly four heads? So, you can already conclude that the Bernoulli distribution is a special case of the binomial distribution with one trial.
Geometric distribution
The geometric distribution is the distribution of the number of Bernoulli trials to get the first success. E.g., if a coin is tossed repeatedly, what is the probability that the first head comes on the fifth trial?
Negative binomial distribution
A general case of the geometric distribution is the negative binomial distribution. It is the distribution of the number of trials needed to get a certain number of successes in repeated independent Bernoulli trials. E.g., if a coin is tossed repeatedly, what is the probability that the third head comes on the tenth trial?
Hypergeometric distribution
The hypergeometric distribution is similar to the binomial distribution but without replacement, or the trials are not independent. E.g., if five cards are drawn from a deck without replacement, what is the probability of getting two spades?
Poisson distribution
It is the distribution of the number of events in a given duration if those are occurring randomly and independently. What is the probability of having exactly three shark attacks on a particular beach this year? The Poisson distribution is approximated to a binomial distribution if the number of trials is large and the probability is small.
Reference
Overview of Some Discrete Probability Distributions: jbstatistics
