Binomial distribution

In Mathematical, a binomial distribution of parameters N and p is a Loi of probability which corresponds to the following experiment:

One renews N time in a way independent a test of Bernoulli of parameter p (experiment random with two exits possible, generally respectively called “success” and “failure”, the probability of a success being p , that of a failure being $q\; =\; (1\; -\; p)$. One then counts the number of successes obtained at the conclusion of N tests and one calls X the Random variable correspondent with this number of successes.

The universe $X\; (\backslash \; Omega)\; ~$ indicates the whole of the natural entireties of 0 with N .

The random variable follows a Loi of probability defined by:

$p\; (K)\; =\; P\; (X\; =\; K)\; =\; \{N\; \backslash \; choose\; K\}\; \backslash ,\; p^k\; q^\; \{n-k\}$

In France, the first term of the member of right-hand side is noted $\backslash \; mathrm\; \{C\}\; \_\; \{N\}\; ^\; \{K\}$. This notation is not recognized internationally.

The symbols $\{N\; \backslash \; choose\; K\}$ and $\backslash \; mathrm\; \{C\}\; \_\; \{N\}\; ^\; \{K\}$ correspond to a number of Combinaison S and are calculated starting from the function Factorielle:

$\{N\; \backslash \; choose\; K\}\; =\; \backslash \; mathrm\; \{C\}\; \_\; \{N\}\; ^\; \{K\}\; =\; \backslash \; frac\; \{N!\}\{K!\; (n-k)!\}$

This law of probability is called the binomial distribution of parameter (N; p) and notes B (N; p) .

Calculation of p (K)

A test of Bernoulli leads to the creation of a universe Ω = {S; E}, (S for Success and E for Failure).

N independent tests of Bernoulli leads to the creation of a Ωⁿ universe made up of N - uplets of elements of Ω, on which can be defined a probability produced. Probability of the possibility (S, S,…, S, E, E,…, E) with K success and N - K failures thus has as a value p^kq^n-k.

More generally, all N - uplet formed of K success and n-k failures will have as a probability p^kq^n-k whatever the order of appearance of the S and the E.

The event “X = K” is made of all N - uplets comprising K success and N - K failures. The Combinatoire makes it possible to determine the number of tuples of this type: there is as much as parts with K elements of a whole with N elements; however each part corresponds to a way of placing K success among N places of the tuple. There is thus $\{N\; \backslash \; choose\; K\}$ tuples, each one having a probability equalizes with p^kq^n-k.

Thus $P\; (X\; =\; K)\; =\; \{N\; \backslash \; choose\; K\}\; \backslash ,\; p^k\; (1-p)\; ^\; \{n-k\}\; =\; \{N\; \backslash \; choose\; K\}\; \backslash ,\; p^k\; q^\; \{n-k\}$.

Hope, variance, standard deviation

X is the sum of N random variable independent according to all it (even) law of Bernoulli of parameter p , taking value 1 in the event of success (probability p ) and 0 in the event of failure (probability (1-p) ); these random variables have for hope p and a variance p (1-p) .

• E (X) is thus the sum of the hopes, either Np

• V (X) is the sum of the variances, or Np (1-p)
• $\backslash \; sigma\_\; \{X\}\; =\; \backslash \; sqrt\; \{Np\; (1-p)\}$

Convergence

For great values of N, the calculation of $\{N\; \backslash \; choose\; K\}\; \backslash ,\; p^k\; q^\; \{n-k\}$ quickly becomes practically impossible, except if one seeks to calculate the logarithm of this expression instead of the expression itself (and on the condition of using the approximation of the factorials by the Formule of Stirling). Two cases are distinguished:

• When N tends towards the infinite one and that p tends towards 0 with Np = has, the binomial distribution converges towards a Loi of Poisson of parameter A. In practice, one replaces the binomial distribution by a law of Poisson as soon as N > 30 and Np < 5.

• When N tends towards the infinite one and that p and Q are of the same order of magnitude, the binomial distribution converges towards a normal Loi of hope Np and variance npq. In practice, one replaces a binomial distribution by a normal law as soon as N > 30, Np > 5 and nq > 5

Law of the great numbers

The binomial distribution, its hope and its variance, as well as the inequality of Bienaymé-Tchebychev make it possible to show a simple version of the Loi of the great numbers.

See too

• Law multinomiale

• Law of probability
• Probability
• Probability (elementary mathematics)
• Random variable elementary
• Walk randomly

External bond

• binomial Table of the Law

• interactive Table of the binomial distribution and graphic interactive

Simple: Binomial distribution

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