probability.tex

\begin{center}
    \textbf{Элементарная теория вероятностей}
\end{center}

$\displaystyle A + B = A + \bar{A}B$;
$\displaystyle \overline{\sum_{i=1}^{n}A_i} = \prod_{i=1}^{n}\bar{A}_i$;
$\displaystyle \sum_{i=1}^{n}\bar{A}_i = \overline{\prod_{i=1}^{n}A_i}$

Попарно несовместны
$\displaystyle P(\sum{A_k}) = \sum{P(A_k)}$;
$\displaystyle P(\sum A_k) = \sum_{k=1}^n (-1)^{k-1}\binom{n}{k}P(\prod_{i=1}^k A_i)$

$\displaystyle P(\prod A_k) = \sum_k P(A_k) - \sum_{k=1}^{n-1}\sum_{i=k+1}^n P(A_k + A_i) +\ldots + (-1)^{n-1}P(\sum_{k=1}^n A_k)$

События независимы в совокупности
$\displaystyle P(\prod A_k) = \prod{P(A_k)}$;
$\displaystyle P(\sum{A_k}) = 1-\prod[1-P(A_k)]$

Условная вероятность
$\displaystyle P(B|A) = \frac{P(AB)}{P(A)}$;
$\displaystyle P\left(\prod_{k=1}^{n}A_k\right) = P(A_1)\cdot P(A_2|A_1)\cdot\ldots\cdot P\left(A_n\left|\right.\prod_{k=1}^{n-1}A_k\right)$

$\displaystyle P(A) = \sum P(H_i)P(A|H_i)$;
$\displaystyle P(H_k|A) = \frac{P(H_k)P(A|H_k)}{\sum P(H_i)P(A|H_i)}$

Бернулли
$\displaystyle P^{(n)}_m = \binom{n}{m}p^mq^{n-m}$;
Пуассон
$\displaystyle P_m = \frac{a^m}{m!}e^{-a}$;
полиномиальный
$\displaystyle P^{(n)}_{m_1\ldots m_k} = \binom{n}{m_1\ldots m_k}p_1^{m_1}\ldots p_k^{m_k}$


\begin{center}
    \textbf{Общая теория вероятностей}
\end{center}

Нормальное распределение
$\displaystyle f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\cfrac{(x-a)^2}{2\sigma^2}}$

$\displaystyle \expect{\xi} =
\sum_{\omega\in\Omega}\xi(\omega)P(\omega) =
\sum_{x\in\xi(\Omega)}xP(\xi=x)$;
$\displaystyle \expect{\xi} = \int\limits_\Omega\xi(\omega)dP$;
$\disper{\xi} = \expect{[(\xi - \expect{\xi})^2]}$

$\displaystyle P(\eta = y | \xi = x) = \frac{P(\eta = y, \xi = x)}{P(\xi = x)}$;
$\displaystyle \expect{[\eta | \xi=x]} = \sum_ly_lP(\eta=y_l|\xi=x_k)$;
$\expect{[\expect{[\eta|\xi]}]} = \expect{\eta}$

$\displaystyle f_{\eta|\xi} =
\frac{f_{\xi,\eta}(x,y)}{\int\limits_{-\infty}^{+\infty}f_{\xi,\eta}(x,u)du}$;
$\displaystyle \expect{[\eta | \xi=x]} =
\int\limits_{-\infty}^{+\infty}yf_{\eta|\xi}(y|\xi=x)dy$;
$\displaystyle \disper{[\eta|\xi=x]} =
\int\limits_{-\infty}^{+\infty}(y-\expect{[\eta|\xi=x]})^2f_{\eta|\xi}(y|x)dy$

$\displaystyle \expect{[g(\xi)|\xi]}=g(\xi),
\expect{[g(\xi)\eta|\xi]} = g(\xi)\expect{[\eta|\xi]}$

$\displaystyle \xi\in\mathbb{N},
G(z) = \sum_{k=0}^\infty P(\xi=k)z^k = \expect{z^\xi},
P(\xi = m) = G_\xi^{(m)}(0), \expect{\xi} = G_\xi'(1),
\disper{\xi} = G_\xi''(1) + G_\xi'(1) - (G_\xi'(1))^2$

$\displaystyle cov(\xi, \eta) =
\expect{[(\xi - \expect{\xi})(\eta - \expect{\eta})]},
\rho(\xi, \eta) =
\frac{cov(\xi, \eta)}
{\sqrt{\disper{\xi}}\sqrt{\disper{\eta}}}$

% TODO неравенства чебышева, ляпунова и проч