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| author | Prefetch | 2022-10-27 20:40:09 +0200 |
|---|---|---|
| committer | Prefetch | 2022-10-27 20:40:09 +0200 |
| commit | 6e70f28ccbd5afc1506f71f013278a9d157ef03a (patch) | |
| tree | a8ca7113917f3e0040d6e5b446e4e41291fd9d3a /source/know/concept/binomial-distribution | |
| parent | bcae81336764eb6c4cdf0f91e2fe632b625dd8b2 (diff) | |
Optimize last images, add proof template, improve CSS
Diffstat (limited to 'source/know/concept/binomial-distribution')
| -rw-r--r-- | source/know/concept/binomial-distribution/index.md | 33 |
1 files changed, 11 insertions, 22 deletions
diff --git a/source/know/concept/binomial-distribution/index.md b/source/know/concept/binomial-distribution/index.md index 1193a93..dc75221 100644 --- a/source/know/concept/binomial-distribution/index.md +++ b/source/know/concept/binomial-distribution/index.md @@ -44,11 +44,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-mean"/> -<label for="proof-mean">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-mean">Proof.</label> + +{% include proof/start.html id="proof-mean" -%} The trick is to treat $$p$$ and $$q$$ as independent until the last moment: $$\begin{aligned} @@ -62,8 +59,8 @@ $$\begin{aligned} \end{aligned}$$ Inserting $$q = 1 - p$$ then gives the desired result. -</div> -</div> +{% include proof/end.html id="proof-mean" %} + Meanwhile, we find the following variance $$\sigma^2$$, with $$\sigma$$ being the standard deviation: @@ -74,12 +71,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-var"/> -<label for="proof-var">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-var">Proof.</label> -We use the same trick to calculate $$\overline{n^2}$$ + +{% include proof/start.html id="proof-var" -%} (the mean squared number of successes): $$\begin{aligned} @@ -106,8 +99,8 @@ $$\begin{aligned} \end{aligned}$$ By inserting $$q = 1 - p$$, we arrive at the desired expression. -</div> -</div> +{% include proof/end.html id="proof-var" %} + As $$N \to \infty$$, the binomial distribution turns into the continuous normal distribution, @@ -119,11 +112,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-normal"/> -<label for="proof-normal">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-normal">Proof.</label> + +{% include proof/start.html id="proof-normal" -%} We take the Taylor expansion of $$\ln\!\big(P_N(n)\big)$$ around the mean $$\mu = Np$$: @@ -211,8 +201,7 @@ $$\begin{aligned} \end{aligned}$$ Taking $$\exp$$ of this expression then yields a normalized Gaussian distribution. -</div> -</div> +{% include proof/end.html id="proof-normal" %} ## References |