From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/binomial-distribution/index.md | 220 +++++++++++++++++++++ 1 file changed, 220 insertions(+) create mode 100644 source/know/concept/binomial-distribution/index.md (limited to 'source/know/concept/binomial-distribution') diff --git a/source/know/concept/binomial-distribution/index.md b/source/know/concept/binomial-distribution/index.md new file mode 100644 index 0000000..c25da3d --- /dev/null +++ b/source/know/concept/binomial-distribution/index.md @@ -0,0 +1,220 @@ +--- +title: "Binomial distribution" +date: 2021-02-26 +categories: +- Statistics +- Mathematics +layout: "concept" +--- + +The **binomial distribution** is a discrete probability distribution +describing a **Bernoulli process**: a set of independent $N$ trials where +each has only two possible outcomes, "success" and "failure", +the former with probability $p$ and the latter with $q = 1 - p$. +The binomial distribution then gives the probability +that $n$ out of the $N$ trials succeed: + +$$\begin{aligned} + \boxed{ + P_N(n) = \binom{N}{n} \: p^n q^{N - n} + } +\end{aligned}$$ + +The first factor is known as the **binomial coefficient**, which describes the +number of microstates (i.e. permutations) that have $n$ successes out of $N$ trials. +These happen to be the coefficients in the polynomial $(a + b)^N$, +and can be read off of Pascal's triangle. +It is defined as follows: + +$$\begin{aligned} + \boxed{ + \binom{N}{n} = \frac{N!}{n! (N - n)!} + } +\end{aligned}$$ + +The remaining factor $p^n (1 - p)^{N - n}$ is then just the +probability of attaining each microstate. + +The expected or mean number of successes $\mu$ after $N$ trials is as follows: + +$$\begin{aligned} + \boxed{ + \mu = N p + } +\end{aligned}$$ + +
+ + + +
+ +Meanwhile, we find the following variance $\sigma^2$, +with $\sigma$ being the standard deviation: + +$$\begin{aligned} + \boxed{ + \sigma^2 = N p q + } +\end{aligned}$$ + +
+ + + +
+ +As $N \to \infty$, the binomial distribution +turns into the continuous normal distribution, +a fact that is sometimes called the **de Moivre-Laplace theorem**: + +$$\begin{aligned} + \boxed{ + \lim_{N \to \infty} P_N(n) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\!\Big(\!-\!\frac{(n - \mu)^2}{2 \sigma^2} \Big) + } +\end{aligned}$$ + +
+ + + +
+ + +## References +1. H. Gould, J. Tobochnik, + *Statistical and thermal physics*, 2nd edition, + Princeton. -- cgit v1.2.3