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Binomial distribution
Probability distribution / From Wikipedia, the free encyclopedia
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In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1-p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.[1]
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PMF |
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CDF |
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in shannons. For nats, use the natural log in the log. | ||
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![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/17/Pascal%27s_triangle%3B_binomial_distribution.svg/640px-Pascal%27s_triangle%3B_binomial_distribution.svg.png)
with n and k as in Pascal's triangle
The probability that a ball in a Galton box with 8 layers (n = 8 ends up in the central bin (k = 4 is 70/256.
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.