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# 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 $b$
### 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
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)`
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+Can be used from Stats: Menu -> Stats -> Cal (bottom) -> Catalog -> Random Bin
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