Probability distribution

The normal distribution, often called the "bell curve"

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 [math]\displaystyle{ X }[/math] is often written as [math]\displaystyle{ f_X(x) }[/math] (or simply [math]\displaystyle{ f(x) }[/math]).^[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	[math]\displaystyle{ 1 }[/math]	[math]\displaystyle{ 2 }[/math]	[math]\displaystyle{ 3 }[/math]	[math]\displaystyle{ 4 }[/math]	[math]\displaystyle{ 5 }[/math]	[math]\displaystyle{ 6 }[/math]
Probability of result	[math]\displaystyle{ \frac {1}{6} }[/math]	[math]\displaystyle{ \frac {1}{6} }[/math]	[math]\displaystyle{ \frac {1}{6} }[/math]	[math]\displaystyle{ \frac {1}{6} }[/math]	[math]\displaystyle{ \frac {1}{6} }[/math]	[math]\displaystyle{ \frac {1}{6} }[/math]

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:

Probability Distribution Media

The left graph shows a probability density function. The right graph shows the cumulative distribution function, for which the value at a equals the area under the probability density curve to the left of a.
The probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice. For example, the figure shows that p(11) = 2/36 = 1/18. The pmf allows the computation of probabilities of events such as P(X > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.
The probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.
The cdf of a discrete probability distribution, ...
... of a continuous probability distribution, ...
... of a distribution which has both a continuous part and a discrete part
One solution for the Rabinovich–Fabrikant equations. What is the probability of observing a state on a certain place of the support (i.e., the red subset)?