Binomial distribution

Binomial distribution

Probability function
Distribution function
Designation
Options	- the number of "tests" - the probability of "success"
Carrier
Probability function
Distribution function
Expected value
Median	one of
Fashion
Dispersion
Asymmetry coefficient
Coefficient of kurtosis
Informational entropy
Generating function of moments
Characteristic function

The binomial distribution in probability theory is the distribution of the number of “successes” in a sequence of independent random experiments, such that the probability of “success” in each of them is constant and equal .

Definition [edit]

Let be - a finite sequence of independent random variables with the same Bernoulli distribution with the parameter that is, with each magnitude takes values ("Success") and ("Failure") with probabilities and respectively. Then a random variable

has a binomial distribution with parameters and . This is written as:

.

Random variable usually interpreted as the number of successes in a series of identical independent Bernoulli tests with probability of success in every test.

The probability function is given by the formula:

Where

- Binomial coefficient.

Distribution function [edit]

The distribution function of the binomial distribution can be written as a sum:

,

Where denotes the largest integer not exceeding the number , or in the form of an incomplete beta function:

.

Moments [edit]

The generating function of moments of the binomial distribution is:

,

from where

,

,

and the variance is a random variable.

.

Properties of the binomial distribution [edit]

• Let be and . Then .
• Let be and . Then .

Relationship with other distributions [edit]

• If a , then, obviously, we obtain the Bernoulli distribution.
• If a large, then by virtue of the central limit theorem where - normal distribution with expectation and variance .
• If a great as well - fixed number, then where - Poisson distribution with parameter .
• If random values and have binomial distributions and accordingly, the conditional distribution of the random variable provided - hypergeometric .

See also [edit]