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author | Prefetch | 2021-02-27 09:57:58 +0100 |
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committer | Prefetch | 2021-02-27 09:57:58 +0100 |
commit | 37d9922b454e738c072d03ad294a07e057fffa50 (patch) | |
tree | 149d3187c726ef5678fa7b0aa61cc22e5d02ddde | |
parent | c705ac1d7dc74709835a8c48fae4a7dd70dc5c49 (diff) |
Improve knowledge base
-rw-r--r-- | content/know/category/optics.md | 9 | ||||
-rw-r--r-- | content/know/category/statistics.md | 9 | ||||
-rw-r--r-- | content/know/concept/binomial-distribution/index.pdc | 193 | ||||
-rw-r--r-- | content/know/concept/dirac-notation/index.pdc | 2 | ||||
-rw-r--r-- | content/know/concept/kramers-kronig-relations/index.pdc | 1 | ||||
-rw-r--r-- | layouts/know/category.html | 2 | ||||
-rw-r--r-- | layouts/know/single.html | 18 |
7 files changed, 233 insertions, 1 deletions
diff --git a/content/know/category/optics.md b/content/know/category/optics.md new file mode 100644 index 0000000..bbfd81b --- /dev/null +++ b/content/know/category/optics.md @@ -0,0 +1,9 @@ +--- +title: "Optics" +firstLetter: "O" +date: 2021-02-26T15:12:43+01:00 +draft: false +layout: "category" +--- + +This page will fill itself. diff --git a/content/know/category/statistics.md b/content/know/category/statistics.md new file mode 100644 index 0000000..ba3e469 --- /dev/null +++ b/content/know/category/statistics.md @@ -0,0 +1,9 @@ +--- +title: "Statistics" +firstLetter: "S" +date: 2021-02-25T21:14:59+01:00 +draft: false +layout: "category" +--- + +This page will fill itself. diff --git a/content/know/concept/binomial-distribution/index.pdc b/content/know/concept/binomial-distribution/index.pdc new file mode 100644 index 0000000..68dec8c --- /dev/null +++ b/content/know/concept/binomial-distribution/index.pdc @@ -0,0 +1,193 @@ +--- +title: "Binomial distribution" +firstLetter: "B" +publishDate: 2021-02-26 +categories: +- Statistics +- Mathematics + +date: 2021-02-25T21:08:52+01:00 +draft: false +markup: pandoc +--- + +# Binomial distribution + +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: + +$$\begin{aligned} + \boxed{ + P_N(n) = \binom{N}{n} \: p^n (1 - p)^{N - n} + } +\end{aligned}$$ + +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. + +To find the mean number of successes $\mu$, +the trick is to treat $p$ and $q$ as independent: + +$$\begin{aligned} + \mu + &= \sum_{n = 0}^N n \binom{N}{n} p^n q^{N - n} + = \sum_{n = 0}^N \binom{N}{n} \Big( p \pdv{(p^n)}{p} \Big) q^{N - n} + \\ + &= p \pdv{p} \sum_{n = 0}^N \binom{N}{n} p^n q^{N - n} + = p \pdv{p} (p + q)^N + = N p (p + q)^{N - 1} +\end{aligned}$$ + +By inserting $q = 1 - p$, we find the following expression for the mean: + +$$\begin{aligned} + \boxed{ + \mu = N p + } +\end{aligned}$$ + +Next, we use the same trick to calculate $\overline{n^2}$ +(the mean of the squared number of successes): + +$$\begin{aligned} + \overline{n^2} + &= \sum_{n = 0}^N n^2 \binom{N}{n} p^n q^{N - n} + = \sum_{n = 0}^N n \binom{N}{n} \Big( p \pdv{p} \Big)^2 p^n q^{N - n} + \\ + &= \Big( p \pdv{p} \Big)^2 \sum_{n = 0}^N \binom{N}{n} p^n q^{N - n} + = \Big( p \pdv{p} \Big)^2 (p + q)^N + \\ + &= N p \pdv{p} p (p + q)^{N - 1} + = N p \big( (p + q)^{N - 1} + (N - 1) p (p + q)^{N - 2} \big) + \\ + &= N p + N^2 p^2 - N p^2 +\end{aligned}$$ + +Using this and the earlier expression for $\mu$, we find the variance $\sigma^2$: + +$$\begin{aligned} + \sigma^2 + &= \overline{n^2} - \mu^2 + = N p + N^2 p^2 - N p^2 - N^2 p^2 + = N p (1 - p) +\end{aligned}$$ + +Once again, by inserting $q = 1 - p$, we find the following expression for the variance: + +$$\begin{aligned} + \boxed{ + \sigma^2 = N p q + } +\end{aligned}$$ + +As $N$ grows to infinity, the binomial distribution +turns into the continuous normal distribution. +We demonstrate this by taking the Taylor expansion of its +natural logarithm $\ln\!\big(P_N(n)\big)$ around the mean $\mu = Np$: + +$$\begin{aligned} + \ln\!\big(P_N(n)\big) + &= \sum_{m = 0}^\infty \frac{(n - \mu)^m}{m!} D_m(\mu) + \quad \mathrm{where} \quad + D_m(n) = \dv[m]{\ln\!\big(P_N(n)\big)}{n} +\end{aligned}$$ + +We use Stirling's approximation to calculate all these factorials: + +$$\begin{aligned} + \ln\!\big(P_N(n)\big) + &= \ln(N!) - \ln(n!) - \ln\!\big((N - n)!\big) + n \ln(p) + (N - n) \ln(q) + \\ + &\approx \ln(N!) - n \big( \ln(n)\!-\!\ln(p)\!-\!1 \big) - (N\!-\!n) \big( \ln(N\!-\!n)\!-\!\ln(q)\!-\!1 \big) +\end{aligned}$$ + +For $D_0(\mu)$, we need to use a stronger version of Stirling's approximation +to get a non-zero result. We take advantage of $N - N p = N q$: + +$$\begin{aligned} + D_0(\mu) + &= \ln(N!) - \ln\!\big((N p)!\big) - \ln\!\big((N q)!\big) + N p \ln(p) + N q \ln(q) + \\ + &= \Big( N \ln(N) - N + \frac{1}{2} \ln(2\pi N) \Big) + - \Big( N p \ln(N p) - N p + \frac{1}{2} \ln(2\pi N p) \Big) \\ + &\qquad - \Big( N q \ln(N q) - N q + \frac{1}{2} \ln(2\pi N q) \Big) + + N p \ln(p) + N q \ln(q) + \\ + &= N \ln(N) - N (p + q) \ln(N) + N (p + q) - N - \frac{1}{2} \ln(2\pi N p q) + \\ + &= - \frac{1}{2} \ln(2\pi N p q) + = \ln\!\Big( \frac{1}{\sqrt{2\pi \sigma^2}} \Big) +\end{aligned}$$ + +Next, we expect that $D_1(\mu) = 0$, because $\mu$ is the maximum. +This is indeed the case: + +$$\begin{aligned} + D_1(n) + &= - \big( \ln(n)\!-\!\ln(p)\!-\!1 \big) + \big( \ln(N\!-\!n)\!-\!\ln(q)\!-\!1 \big) - 1 + 1 + \\ + &= - \ln(n) + \ln(N - n) + \ln(p) - \ln(q) + \\ + D_1(\mu) + &= \ln(N q) - \ln(N p) + \ln(p) - \ln(q) + = \ln(N p q) - \ln(N p q) + = 0 +\end{aligned}$$ + +For the same reason, we expect that $D_2(\mu)$ is negative +We find the following expression: + +$$\begin{aligned} + D_2(n) + &= - \frac{1}{n} - \frac{1}{N - n} + \qquad + D_2(\mu) + = - \frac{1}{Np} - \frac{1}{Nq} + = - \frac{p + q}{N p q} + = - \frac{1}{\sigma^2} +\end{aligned}$$ + +The higher-order derivatives tend to zero for large $N$, so we discard them: + +$$\begin{aligned} + D_3(n) + = \frac{1}{n^2} - \frac{1}{(N - n)^2} + \qquad + D_4(n) + = - \frac{2}{n^3} - \frac{2}{(N - n)^3} + \qquad + \cdots +\end{aligned}$$ + +Putting everything together, for large $N$, +the Taylor series approximately becomes: + +$$\begin{aligned} + \ln\!\big(P_N(n)\big) + \approx D_0(\mu) + \frac{(n - \mu)^2}{2} D_2(\mu) + = \ln\!\Big( \frac{1}{\sqrt{2\pi \sigma^2}} \Big) - \frac{(n - \mu)^2}{2 \sigma^2} +\end{aligned}$$ + +Thus, as $N$ goes to infinity, the binomial distribution becomes a Gaussian: + +$$\begin{aligned} + \boxed{ + \lim_{N \to \infty} P_N(n) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\!\Big(\!-\!\frac{(n - \mu)^2}{2 \sigma^2} \Big) + } +\end{aligned}$$ diff --git a/content/know/concept/dirac-notation/index.pdc b/content/know/concept/dirac-notation/index.pdc index f624574..999c90c 100644 --- a/content/know/concept/dirac-notation/index.pdc +++ b/content/know/concept/dirac-notation/index.pdc @@ -13,7 +13,7 @@ markup: pandoc # Dirac notation -**Dirac notation** is a notation to do calculations in a Hilbert space +**Dirac notation** is a notation to do calculations in a [Hilbert space](/know/concept/hilbert-space/) without needing to worry about the space's representation. It is basically the *lingua franca* of quantum mechanics. diff --git a/content/know/concept/kramers-kronig-relations/index.pdc b/content/know/concept/kramers-kronig-relations/index.pdc index 1c2977e..1d29f96 100644 --- a/content/know/concept/kramers-kronig-relations/index.pdc +++ b/content/know/concept/kramers-kronig-relations/index.pdc @@ -5,6 +5,7 @@ publishDate: 2021-02-25 categories: - Mathematics - Physics +- Optics date: 2021-02-25T15:20:24+01:00 draft: false diff --git a/layouts/know/category.html b/layouts/know/category.html index 3dc8c7a..a672ea1 100644 --- a/layouts/know/category.html +++ b/layouts/know/category.html @@ -20,7 +20,9 @@ {{ end }} </div> +<p> This is an alphabetical list of the concepts in this category. +</p> {{ range $relevant.GroupByParam "firstLetter" }} <h2 id="{{ .Key }}">{{ .Key }}</h2> diff --git a/layouts/know/single.html b/layouts/know/single.html index d14a83c..fe1197b 100644 --- a/layouts/know/single.html +++ b/layouts/know/single.html @@ -4,10 +4,28 @@ {{ partial "head.html" . }} {{ partial "mathjax.html" . }} <title>{{ .Title }} | {{ .Site.Title }}</title> +<style> +@media (prefers-color-scheme: dark) { +img {filter:invert(100%);max-width:100%;height:auto;} +} +</style> </head> <body> {{ partial "navigation.html" . }} + +<p> +Categories: +{{ $len := len .Params.categories }} +{{ $idx := 0 }} +{{ range sort .Params.categories }} + {{ $idx = add $idx 1 }} + {{ $url := replace . " " "-" | lower }} + <a href="/know/category/{{ $url }}/">{{ . }}</a>{{ if eq $idx $len }}.{{ else }},{{end}} +{{ end }} +</p> + {{ .Content }} + {{ partial "footer.html" . }} </body> </html> |