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--- math [2020/09/26 13:26] – created admin
+++ math [2023/08/12 04:40] (current) – [pages in this wiki] raju
@@ Line 1: / Line 1: @@
-==== simplifications ====
+==== correlation ====
+t-statistic for correlation, $r$
+\[
+t = r \sqrt{\frac{n-2}{1-r^2}}
+\]
+The reverse equation is
 \[
-\sec x + \tan x = \tan \left( \frac{x}{2} + \frac{\pi}{4} \right) = \sqrt{\frac{1 + \sin x}{1 - \sin x}}
+r = \frac{t}{\sqrt{t^2 + n-2}}
 \]
+Derivation of the reverse equation:
+\begin{align*}
+& t = r \sqrt{\frac{n-2}{1-r^2}} & \\
+& \frac{t^2}{r^2} = \frac{n-2}{1-r^2} & \\
+& \frac{t^2}{n-2} = \frac{r^2}{1-r^2} & \\
+& \frac{t^2}{t^2 + n-2} = r^2 & \\
+& r = \sqrt{\frac{t^2}{t^2 + n-2}} &
+\end{align*}
+See also:
+  * https://stackoverflow.com/questions/2632628/left-align-block-of-equations - shows a trick to left align an equation array
+==== sec(x) + tan(x) ====
+\begin{align*}
+\sec x + \tan x & = \tan \left( \frac{x}{2} + \frac{\pi}{4} \right) \\
+& = \sqrt{\frac{1 + \sin x}{1 - \sin x}}
+\end{align*}
+I came across this simplification while reading https://liorsinai.github.io/mathematics/2020/08/27/secant-mercator.html which talks about the integral of the secant which in turn has applications in Mercator map.
+==== pages in this wiki ====
+  * [[fibonacci sequence]]
+  * [[Leibniz integral rule]]
+==== links I came across ====
+  * What is the formula for sum of the p-th powers of the first n positive integers?
+    * Use Faulhaber's formula in described in https://en.wikipedia.org/wiki/Faulhaber%27s_formula
+    * The formula involves binomial coefficients and Bernoulli numbers.
+    * See also: https://www.johndcook.com/blog/2016/12/31/sums-of-consecutive-powers/
+==== pages with math outside of this wiki ====
+use cases | to copy paste syntax
+  * https://liorsinai.github.io/mathematics/2020/08/27/secant-mercator.html
+  * https://en.wikipedia.org/wiki/Fibonacci_sequence
+  * https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors