1--- 2header-includes: 3- \usepackage{amsmath} 4-\newcommand*{\perm}[2]{{}^{#1}\!P_{#2}}% 5-\newcommand*{\comb}[2]{{}^{#1}C_{#2}}% 6--- 7 8# Sampling and Distributions 9 10**Population** - set of all eligible members 11**Sample** - subset of population, may be representative of population 12**Random sample** - every element of population has equal chance of selection (exactly representative) 13**Population mean $\mu$** - mean of all values of one attribute in population. $\mu = {\Sigma \text{values} \over |\text{population}|}$ 14**Sample mean $\overline{x}$** - mean of all values in a sample. Random variable. 15**Population proportion $p$** - proportion of defectives in population (100% certainty) 16**Sample proportion $\hat{p}$** - proportion of defectives in one sample (varies between samples) 17**Discrete random variable** - countable number of distinct values 18 19$$\sum \Pr(n)=1$$ 20 21### Hypergeometric distribution 22 23$$\Pr(X=x) = {{\binom Dx \binom {N-d}{N-x}} \over{\binom Nn}}$$ 24 25where $N$ is the size of population, $n$ is the size of the sample, and $D$ is the number of defectives in population 26 27On CAS: Interactive -> Distribution -> HyperGeoPdf 28 29### Generating random numbers 30Catalog -> `rand(a,b)` generates a random number between $a$ and $b$ 31`randlist(n,a,b)` generates $n$ random numbers between $a$ and $b$ 32 33### Combinations 34 35$$^n C_r = {n! \over {r!\cdot (n-r)!}} = \binom nr$$ 36 37$n$ is population, $r$ is sample (i.e. no of combinations of $r$ in $n$) 38CAS: Advanced -> `nCr(n,r)` $= ^nC_r$ 39 40### Binomial distributions 41 42with replacement. 43 44probability of achieving $x$ successes in $n$ trials for random variable $X$: 45 46$$\Pr(X=x)=\binom nx p^x (1-p)^{n-x} \quad \text{for }x = 0,1,2, \dots, n$$ 47 48where $p$ = probability of success on each trial 49 50##### Simulate on CAS: 51 52`randBin(sample size, p^, no of samples)` 53 54Can be used from Stats: Menu -> Stats -> Cal (bottom) -> Catalog -> Random Bin