Normalisation (probability)

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

In mathematical probability equations, which are used in nearly all branches of science, a normalization constant (or function) is often used to ensure that the sum of all probabilities totals one, or

∑Pi=1

Probability distributions can be divided into two main groups: discrete probability distributions and continuous probability distributions.

## Discrete Probability Distributions[edit]

Discrete probability distributions are used throughout gaming theory. Consider the simple example of rolling a pair of six-sided dice. Summing up the total roll of the dice yields the following possibilities:

Total (i)| Possible outcomes (Die1,Die2)| Occurrences (ni)

|
|

2|  (1,1) |  1
3|  (1,2), (2,1) |  2
4|  (1,3), (3,1), (2,2) |  3
5|  (1,4), (4,1), (2,3), (3,2) |  4
6|  (1,5), (5,1), (2,4), (4,2), (3,3) |  5
7|  (1,6), (6,1), (2,5), (5,2), (3,4), (4,3) |  6
8|  (2,6), (6,2), (5,3), (3,5), (4,4) |  5
9|  (3,6), (6,3), (4,5), (5,4) |  4
10|  (4,6), (6,4), (5,5) |  3
11|  (5,6), (6,5) |  2
12|  (6,6) |  1

Since the probability of any particular outcome is proportional to the number of ways it can occur

∑Pi=∑cini=∑Nni=1

where ci is a coefficient of probability for outcome i. Assuming the dice are symmetrical we assume all values of ci are equal and their sum equals 1.

Solving for N yields 1/36, the number of possible outcomes, so that the probability of total = i occurring are

Pi=(136)ni, and the sum of all probabilities is one

∑Pi=(136)(1+2+3+4+5+6+5+4+3+2+1)=3636=1

## Continuous probability distributions[edit]

In most scientific equations, probability functions are continuous functions, and the probability coefficients are sometimes functions rather than constants. For example, the zeta distribution with parameter s assigns probability proportional to 1/ns to the integer n: the normalizing factor is then the value of the Riemann zeta function

ζ(s)=∑n=1∞1ns.