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 $$(z_{1} <= a') \land (z_{n} >= b') \land \forall k ( z_{k} < z_{k+1} <= z_{k} + \zeta)$$
 * 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.