Discrete Probability Distribution

Probability Mass Function

Denoted by . Gives the probability that a discrete random variable is exactly equal to some value .

Parameters

Mean

Here:

• represents each possible value of
• is the probability of observing that value

Variance

An equivalent computational formula is:

Cumulative Distribution Function

Example

Consider a discrete random variable with the following probability distribution:

X	P(X)
1	0.2
2	0.3
3	0.4
4	0.1

The mean would be: .

The variance would be: .

Types

Uniform distribution

A type of discrete probability distribution where all outcomes are equally likely. If a random variable can take distinct values, each value has a probability of .

Bernoulli distribution

A type of discrete probability distribution where there are only two possible outcomes, often referred to as “success” and “failure”. If a random variable can take values 0 or 1, with and , then follows a Bernoulli distribution.

PMF of Bernoulli distribution:

Here:

• is the probability of success

Binomial distribution

When an experiment consists of independent repeated Bernoulli trials, the experiment is said to be a binomial experiment. The number of successes () follows a binomial distribution. And has a PMF:

Here:

• is the number of trials.
• is the probability of success.

Assumptions made in binomial distribution:

• Each trial is independent.
• Each trial has only two possible outcomes.
• The probability of success is constant across trials.
• The number of trials is fixed.

For a large sample of a binomial distribution:

Poisson distribution

If is the number of independent successes occur within a fixed time interval, then is said to follow a Poisson distribution. And has a PMF:

Here:

• - average number of successes occurred within the fixed time interval

If where , and is independent then: