From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- source/know/concept/binomial-distribution/index.md | 50 +++++++++++----------- 1 file changed, 25 insertions(+), 25 deletions(-) (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 index 14ba4cb..1193a93 100644 --- a/source/know/concept/binomial-distribution/index.md +++ b/source/know/concept/binomial-distribution/index.md @@ -9,11 +9,11 @@ layout: "concept" --- The **binomial distribution** is a discrete probability distribution -describing a **Bernoulli process**: a set of independent $N$ trials where +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 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: +that $$n$$ out of the $$N$$ trials succeed: $$\begin{aligned} \boxed{ @@ -22,8 +22,8 @@ $$\begin{aligned} \end{aligned}$$ The first factor is known as the **binomial coefficient**, which describes the -number of microstates (i.e. permutations) that have $n$ successes out of $N$ trials. -These happen to be the coefficients in the polynomial $(a + b)^N$, +number of microstates (i.e. permutations) that have $$n$$ successes out of $$N$$ trials. +These happen to be the coefficients in the polynomial $$(a + b)^N$$, and can be read off of Pascal's triangle. It is defined as follows: @@ -33,10 +33,10 @@ $$\begin{aligned} } \end{aligned}$$ -The remaining factor $p^n (1 - p)^{N - n}$ is then just the +The remaining factor $$p^n (1 - p)^{N - n}$$ is then just the probability of attaining each microstate. -The expected or mean number of successes $\mu$ after $N$ trials is as follows: +The expected or mean number of successes $$\mu$$ after $$N$$ trials is as follows: $$\begin{aligned} \boxed{ @@ -49,7 +49,7 @@ $$\begin{aligned} -Meanwhile, we find the following variance $\sigma^2$, -with $\sigma$ being the standard deviation: +Meanwhile, we find the following variance $$\sigma^2$$, +with $$\sigma$$ being the standard deviation: $$\begin{aligned} \boxed{ @@ -79,7 +79,7 @@ $$\begin{aligned} -As $N \to \infty$, the binomial distribution +As $$N \to \infty$$, the binomial distribution turns into the continuous normal distribution, a fact that is sometimes called the **de Moivre-Laplace theorem**: @@ -124,8 +124,8 @@ $$\begin{aligned} -- cgit v1.2.3