Probability distribution

Probability distribution is a term from mathematics. Suppose there are many events with random outcomes. A probability distribution is the theoretical counterpart to the frequency distribution. A frequency distribution simply shows how many times a certain event occurred. A probability distribution says how many times it should have occurred in the long run (that is, its probability). The probability distribution of a random variable ${\displaystyle X}$ is often written as ${\displaystyle f_{X}(x)}$ (or simply ${\displaystyle f(x)}$).^[1][2] Such a distribution can either be discrete, taking a discrete (or countable) amount of values, or continuous, taking an uncountable amount of values (as from a continuous interval).^[3]

As an example, the probability distribution for a single roll of a normal 6-sided dice can be presented by:

Probability distribution for a dice roll event

Result	${\displaystyle 1}$	${\displaystyle 2}$	${\displaystyle 3}$	${\displaystyle 4}$	${\displaystyle 5}$	${\displaystyle 6}$
Probability of result	${\displaystyle {\frac {1}{6}}}$	${\displaystyle {\frac {1}{6}}}$	${\displaystyle {\frac {1}{6}}}$	${\displaystyle {\frac {1}{6}}}$	${\displaystyle {\frac {1}{6}}}$	${\displaystyle {\frac {1}{6}}}$

where result is the outcome of the dice roll, and the probability shows the chances of that result occurring. If we roll a dice 60 times, then in the long run, we should expect to have each side appear 10 times on average.

There are different probability distributions.^[4] Each of them has its use, its benefits and its drawbacks. Some common probability distributions include:

• Binomial distribution
• Cauchy distribution
• Chi-square distribution
• Exponential distribution
• Gumbel distribution
• Normal distribution
• Poisson distribution
• Student's t-distribution

1. ↑ "List of Probability and Statistics Symbols". Math Vault. 2020-04-26. Retrieved 2020-09-11.
2. ↑ Bourne, Murray. "11. Probability Distributions - Concepts". www.intmath.com. Retrieved 2020-09-11.
3. ↑ "1.3.6.1. What is a Probability Distribution". www.itl.nist.gov. Retrieved 2020-09-11.
4. ↑ "Normal Distribution - easily explained! | Data Basecamp". 2021-11-26. Retrieved 2023-05-29.