Discrete Distributions
In mathematics, many different statistical distributions model varying situations. These distributions fall into two categories, discrete and continuous. Discrete distributions involve situations where the output are distinct and separate from other outcomes. Continuous distributions are situations like height, where a person’s height can be calculated to many decimal places and found to be different from almost every other person.
The most basic type of discrete distribution is the Bernoulli distribution. Named after Jacob Bernoulli, a member of the Bernoulli family of mathematicians, the Bernoulli distributions model situations with two possible outcomes. It is modeled by the probability mass function (pmf), f(x) = p^x*(1-p)^(1-x), where x can only be 0 and 1 and p is the probability of success. The expected value is p.
Based on the Bernoulli distribution, the binomial distributions model situations with repeated Bernoulli trials. The pmf is the same as the Bernoulli distribution multiplied by n choose x, meaning you do not care what order your successes happen in. The expected value is n*p, where n is the number of Bernoulli trials.
Another very basic distribution is the uniform distribution where all outcomes have the same probability. The pmf is 1/m where m is the number of possible outcomes. The expected value is (m+1)/2. An example of the uniform distribution is a single dice roll.
The geometric distribution is a discrete distribution that models situations where you continue to perform a trial until you succeed. For example, winning streaks in sports or flipping a coin until you it lands on tails. The pmf is f(x) = (1-p)^(x-1)*p, and the expected value is 1/p.
The negative binomial distribution is just a more general form of the geometric distribution. The geometric distribution models the number of failures before a single success. The negative binomial distribution models the number of failures before any number of successes. The pmf is
f(x) = (x-1)C(r-1)*p^r*(1-p)^(x-r). Where r is the number of successes you are looking for.
The hypergeometric distribution sounds like it has to do with the geometric distribution, but they have very little in common. The hypergeometric distribution models the probability of drawing an item without replacement a given number of times. The pmf is given by the picture underneath.
The final discrete distribution is the Poisson distribution. This distribution models the probability of a given number of successes in an interval of time or space. For example it would tell you the probability of a team scoring 2 goals in the first 20 minutes of a soccer match. Or it would tell you the probability of encountering a planet if you fly one lightyear in a spaceship. The pmf is modeled by f(x) = (λ^x*e^(-λ))/x!. Interestingly the expected value and the variance are both λ.
These are many of the discrete distributions statisticians use to model situations that they might come accross.
