1# Sampling and Distributions 2 3**Population** - set of all eligible members 4**Sample** - subset of population, may be representative of population 5**Random sample** - every element of population has equal chance of selection 6**Population proportion $p$** - proportion of individuals in population with an attribute 7**Sample proportion $\^p$** - 8**Discrete random variable** - countable number of distinct values 9 10$$\sum \Pr(n)=1$$ 11 12### Hypergeometric distribution 13 14$$\Pr(X=x) = {{{\begin{Bmatrix} 15 D \\ 16 x \\ 17 \end{Bmatrix}}\begin{Bmatrix} {N-D} \\ {n-x} \end{Bmatrix} }\over\begin{Bmatrix}N \\ n \end{Bmatrix}}$$ 18 19 20### Generating random numbers 21Catalog -> `rand(a,b)` generates a random number between $a$ and $b$ 22`randlist(n,a,b)` generates $n$ random numbers between $a$ and $b$ 23 24### Combinations 25 26CAS: Advanced -> `nCr(n,r)` $= ^nC_r$ 27 28### Binomial distributions 29 30with replacement. 31 32probability of achieving $x$ successes in $n$ trials for random variable $X$: 33 34$$\Pr(X=x)=\begin{Bmatrix} n \\ x \end{Bmatrix} p^x (1-p)^{n-x} \quad \text{for }x = 0,1,2, \dots, n$$ 35 36where $p$ = probability of success on each trial 37 38#### on CAS: 39 40`randBin(sample size, p^, no of samples)`