From 8f2bff49410cc88fc2b3babd03525cdba6043c4c Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Thu, 3 Jan 2019 13:54:29 +1100
Subject: [PATCH] finish prelim & sampling from 2018

---
 spec/prelim.md   | 16 +++++++++++++++-
 spec/sampling.md | 40 ++++++++++++++++++++++++++++++++++++++++
 2 files changed, 55 insertions(+), 1 deletion(-)
 create mode 100644 spec/sampling.md

diff --git a/spec/prelim.md b/spec/prelim.md
index c01fe95..d158bbb 100644
--- a/spec/prelim.md
+++ b/spec/prelim.md
@@ -85,4 +85,18 @@ ${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$ and ${(y-k)^2 \over b^2} - {(x-h
 
 ## Modulus function
 
-$$|x|=\sqrt{x^2}$$
\ No newline at end of file
+$$|x|=\sqrt{x^2}$$
+
+## Parametric equations
+
+### Circles
+$$\[\begin{cases}
+        x=a\cos t\\
+        y=a\sin t
+    \end{cases}
+\text{where radius} =a$$
+
+To convert to cartesian, factorise and use $\cos^2 x + \sin^2 x=1$
+
+$\cos^2 t + \sin^2 t = 1$  
+$\implies {\cos^2 \over \sin^2 t} + {\sin^2 t \over sin^2 t} = {1 \over \sin^2 t} \implies \csc^2 t - \cot^2 t$
diff --git a/spec/sampling.md b/spec/sampling.md
new file mode 100644
index 0000000..2205b4b
--- /dev/null
+++ b/spec/sampling.md
@@ -0,0 +1,40 @@
+# Sampling and Distributions
+
+**Population** - set of all eligible members  
+**Sample** - subset of population, may be representative of population  
+**Random sample** - every element of population has equal chance of selection  
+**Population proportion $p$** - proportion of individuals in population with an attribute  
+**Sample proportion $\^p$** -
+**Discrete random variable** - countable number of distinct values
+
+$$\sum \Pr(n)=1$$
+
+### Hypergeometric distribution
+
+$$\Pr(X=x) = {{{\begin{Bmatrix}
+  D \\
+  x \\
+  \end{Bmatrix}}\begin{Bmatrix} {N-D} \\ {n-x} \end{Bmatrix} }\over\begin{Bmatrix}N \\ n \end{Bmatrix}}$$
+
+
+### Generating random numbers
+Catalog -> `rand(a,b)` generates a random number between $a$ and $b$  
+`randlist(n,a,b)` generates $n$ random numbers between $a$ and $b$
+
+### Combinations
+
+CAS: Advanced -> `nCr(n,r)` $= ^nC_r$
+
+### Binomial distributions
+
+with replacement.
+
+probability of achieving $x$ successes in $n$ trials for random variable $X$:
+
+$$\Pr(X=x)=\begin{Bmatrix} n \\ x \end{Bmatrix} p^x (1-p)^{n-x} \quad \text{for }x = 0,1,2, \dots, n$$
+
+where $p$ = probability of success on each trial
+
+#### on CAS:
+
+`randBin(sample size, p^, no of samples)`
\ No newline at end of file
-- 
2.47.1