User:Grover cleveland/Statistics

1. Def: Let $\Omega$ be a set (of "atomic outcomes"). A set $E$ is an event iff $E\subseteq \Omega$

2. Def: Let $P$ be a (partial) function from, $\mathbf {P} (\Omega )$ to $[0,1]$. $P$ is a probability function iff it satisfies the following three conditions:

2.1$P(\Omega )=1$

2.2 For any events $A,B$, if $A\subseteq B$, and both $P(A)$ and $P(B)$ are defined, then $P(B-A)=P(B)-P(A)$

2.3 For any countable sequence $A_{1}...A_{n}...$ of pairwise disjoint events, such that all $P(A_{m})$ are defined for $1\leq m\leq n$, then $P(\bigcup \{A_{1}...A_{n}...\})=\Sigma P(A_{1})...P(A_{n})...$

3. $P(\emptyset )=0$ (from 2.1 and 2.2)

4. For any events $A,B$, then if any three of$P(A),P(B),P(A\cup B)$ and $P(A\cap B)$ are defined, then all four must be, and $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ (from 2.2 and 2.3)

5. Def: given a $P$, two events $A,B$ are independent iff $P(A\cap B)=P(A)\cdot P(B)$.

6. Def: Given any events $A,B$, if $P(B)>0$, then $P(A|B)$ ("probability of A given B"), is defined as $P(A\cap B)/P(B)$.

6.1 If $A,B$ are independent, and $0<P(B)<1$, then $P(A|B)=P(A|\Omega -B)$.

7. Def: "x is a real variable" iff $\Omega =\{x=r|r\in \mathbb {R} \}$ and $P(x<r)$ is defined for all real $r$.

7.1 If x is a real variable, then $P(r_{1}\leq x<r_{2})$ is defined for all real $r_{1},r_{2}$ (from 7).

8. Def: if x is a real variable, then $y$ is the expected value of $x$ ($y=\operatorname {E} [x]$) iff

for any $\epsilon >0$ and any real numbers $a,b$

there exists a $\zeta >0$ and real numbers $a'\leq a$ and $b'\geq b$ such that

for any nonzero $n\in \mathbb {N}$ and any sequence$\{z_{1}...z_{n}\}$ such that $(z_{1}<=a')\land (z_{n}>=b')\land \forall k(z_{k}<z_{k+1}<=z_{k}+\zeta )$

$|\sum _{k=1}^{k=n-1}(z_{k}\centerdot (P(z_{k}\leq x<z_{k+1}))-y|<=\epsilon$

. 9. Def: if the expected value of $x$ is defined, then the variance of $x$ is the expected value of $(x-\operatorname {E} [x])^{2}$. The standard deviation is the square root of the variance.