From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Mon, 28 Jan 2019 06:45:12 +0000 (+1100)
Subject: add to sampling notes (combiantorics & permutations)
X-Git-Tag: yr12~277
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/ad1a07cfa82a26f0609e811c32b1847e38c42800

add to sampling notes (combiantorics & permutations)
---

diff --git a/spec/sampling.md b/spec/sampling.md
index 2205b4b..6ba0e05 100644
--- a/spec/sampling.md
+++ b/spec/sampling.md
@@ -1,21 +1,30 @@
+---
+header-includes:
+- \usepackage{amsmath}
+- \newcommand*{\perm}[2]{{}^{#1}\!P_{#2}}%
+- \newcommand*{\comb}[2]{{}^{#1}C_{#2}}%
+---
+
 # 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 $\^p$** -
+**Random sample** - every element of population has equal chance of selection (exactly representative)  
+**Population mean $\mu$** - mean of all values of one attribute in population. $\mu = {\Sigma \text{values} \over |\text{population}|}$  
+**Sample mean $\overline{x}$** - mean of all values in a sample. Random variable.  
+**Population proportion $p$** - proportion of defectives in population (100% certainty)  
+**Sample proportion $\hat{p}$** - proportion of defectives in one sample (varies between samples)   
 **Discrete random variable** - countable number of distinct values
 
 $$\sum \Pr(n)=1$$
 
 ### Hypergeometric distribution
 
-$$\Pr(X=x) = {{{\begin{Bmatrix}
-  D \\
-  x \\
-  \end{Bmatrix}}\begin{Bmatrix} {N-D} \\ {n-x} \end{Bmatrix} }\over\begin{Bmatrix}N \\ n \end{Bmatrix}}$$
+$$\Pr(X=x) = {{\binom Dx \binom {N-d}{N-x}} \over{\binom Nn}}$$
+
+where $N$ is the size of population, $n$ is the size of the sample, and $D$ is the number of defectives in population
 
+On CAS: Interactive -> Distribution -> HyperGeoPdf
 
 ### Generating random numbers
 Catalog -> `rand(a,b)` generates a random number between $a$ and $b$  
@@ -23,7 +32,10 @@ Catalog -> `rand(a,b)` generates a random number between $a$ and $b$
 
 ### Combinations
 
-CAS: Advanced -> `nCr(n,r)` $= ^nC_r$
+$$^n C_r = {n! \over {r!\cdot (n-r)!}} = \binom nr$$
+
+$n$ is population, $r$ is sample (i.e. no of combinations of $r$ in $n$)
+CAS: Advanced -> `nCr(n,r)` $= ^nC_r$  
 
 ### Binomial distributions
 
@@ -31,10 +43,12 @@ with replacement.
 
 probability of achieving $x$ successes in $n$ trials for random variable $X$:
 
-$$\Pr(X=x)=\begin{Bmatrix} n \\ x \end{Bmatrix} p^x (1-p)^{n-x} \quad \text{for }x = 0,1,2, \dots, n$$
+$$\Pr(X=x)=\binom nx p^x (1-p)^{n-x} \quad \text{for }x = 0,1,2, \dots, n$$
 
 where $p$ = probability of success on each trial
 
-#### on CAS:
+##### Simulate on CAS:
+
+`randBin(sample size, p^, no of samples)`
 
-`randBin(sample size, p^, no of samples)`
\ No newline at end of file
+Can be used from Stats: Menu -> Stats -> Cal (bottom) -> Catalog -> Random Bin  
\ No newline at end of file