User:Amyqz/sandbox
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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 experiment either successes with probability or fails with probability
. 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.
Probability mass function ![]() | |||
Cumulative distribution function ![]() | |||
Notation | B(n, p) | ||
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Parameters |
n ∈ N0 — number of trials p ∈ [0,1] — success probability in each trial | ||
Support | k ∈ { 0, …, n } — number of successes | ||
PMF |
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CDF |
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Mean |
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Median |
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Mode |
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Variance |
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Skewness |
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Excess kurtosis |
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Entropy |
in shannons. For nats, use the natural log in the log. | ||
MGF |
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CF |
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PGF |
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Fisher information |
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![Thumb image](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
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.