AlexTheGer

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Lienare Algebra & Vektoren

$Pr:\Omega \rightarrow [0,1]$

$\sum _{a\in A}Pr(a)=1$

$A\subseteq \Omega$

$Pr:{\mathcal {P}}(\Omega )\rightarrow [0,1]$

$Pr(A)=\sum _{a\in A}Pr(a)$

$P(\Omega )=2^{\Omega }$

${\frac {1}{6}}$

$\neg$

$\cup$ $\in$

$\cap$

$\subseteq$

$\varnothing$

$Pr(B)={5 \choose 3}\cdot {\frac {1}{2^{5}}}$

$Pr(A)={5 \over 6}^{k-1}\cdot {1 \over 6}={5^{k-1} \over 6^{k}}$

$A\in {\mathcal {F}}\Rightarrow {\overline {A}}\in {\mathcal {F}}$

$A_{1},A_{2},...\in {\mathcal {F}}\Rightarrow \bigcup _{i=1}^{\infty }A_{i}\in {\mathcal {F}}$

${\overline {A}}$

$A_{i}\in {\mathcal {F}}$

$Pr\left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }Pr(A_{i})$

${\mathcal {F}}\subseteq P(\Omega )$

$Pr(A|B)={\frac {Pr(A\cap B)}{Pr(B)}}$

$Pr(A)=\sum _{i}Pr(A|B_{i})\cdot Pr(B_{i})$

$E(X)=\sum _{w\in \Omega }X(w)\cdot Pr({w})$

$ImX=\{x\in R|\exists a\in \Omega \quad X(a)=x\}$

$\forall x\in R\quad X^{-1}(x)=\{a\in \Omega |x(a)=x\}\in {\mathcal {F}}$

$E(X)=\sum _{x\in Im(X)}x\cdot Pr(X^{-1}(x))$

$X^{-1}(T)\in {\mathcal {F}}$

$X^{-1}(T)=\bigcup _{x\in (T\cap ImX)}X^{-1}(T)$

$Pr(X^{-1}(x))=Pr_{X}(x)=Pr(X=x)\$

$Pr_{X}(k)={n \choose k}\cdot p^{k}\cdot q^{n-k}$ $k\in \{0,1,...,n\}$

$Pr_{X}(k)=p\cdot q^{k-1}$ $k\in \mathbb {N} ^{+}$

$\mathbb {N} ^{+}$

${\frac {1}{6}}\left({\frac {5}{6}}\right)^{n-1}$

$Pr_{X}(k)={\frac {1}{k!}}\lambda ^{k}e^{-\lambda }$

$\{a\in \Omega |X(a)\leq x\}\in {\mathcal {F}}$

$F_{X}(x)=Pr(\{a\in \Omega |X(a)\leq x\})=Pr(X\leq x)$

$F_{X}(x)=\sum _{y\leq x}Pr_{X}(y)$

$x\leq y\Rightarrow F(x)\leq F(y)$

$\lim _{x\to -\infty }F(x)=0$ $\lim _{x\to \infty }F(x)=1$

$\forall x\in \mathbb {R} \lim _{h\to 0+}F(x+h)=F(x)$

$F_{X}(x)=\int _{-\infty }^{x}f(t)\mathrm {d} t$

$E(X)=\sum _{x\in \mathrm {Im} X}x\cdot Pr_{X}(x)=\sum _{x\in \mathrm {Im} x}x\cdot Pr(\{a\in \Omega |X(a)=x\})$

$E(X)=\int _{-\infty }^{\infty }x\cdot f(x)\mathrm {d} x$

$X=X_{1}+X_{2}+X_{3}+...+X_{n}\$

$E(X)=E(X_{1})+E(X_{2})+E(X_{3})+...+E(X_{n})=n\cdot p$

$Pr_{X}(k)=(e-p)^{k-1}\cdot p=q^{k-1}\cdot p$

$E(X)=\sum _{k=1}^{\infty }k\cdot q^{k-1}\cdot p=...=1\cdot {\frac {1}{1-q}}={\frac {1}{p}}$

$E(X)=\sum _{k=0}^{\infty }k\cdot {\frac {1}{k!}}\cdot \lambda ^{k}\cdot e^{-\lambda }=...=\lambda \cdot e^{\lambda }\cdot e^{-\lambda }=\lambda$

${\frac {1}{b-a}}$

$E(X)=\int _{-\infty }^{\infty }x\cdot f(x)\mathrm {d} x=\int _{a}^{b}x\cdot {\frac {1}{b-a}}\mathrm {d} x=...={\frac {a+b}{2}}$

$X:\Omega \rightarrow \mathbb {R} ^{\geq 0}$

$Pr(X\geq t)\leq {\frac {E(X)}{t}}$ $Pr(X\geq \alpha E(X))\leq {\frac {1}{\alpha }}$

$t=2{,}0m\quad Pr(X\geq 2{,}0)\leq {\frac {E(X)}{t}}={\frac {1{,}7}{2{,}0}}=0{,}85$

$t=0{,}5m\quad Pr(Y\geq 0{,}5)\leq {\frac {0{,}2}{0{,}5}}=0{,}4$

$E(X^{2})=\sum _{x\in ImX}x^{2}\cdot Pr(X=x)$

$E(X)=\int _{-\infty }^{\infty }x\cdot f_{X}(x)\mathrm {d} x$

$E(Y)=\int _{-\infty }^{\infty }y\cdot f_{Y}(y)\mathrm {d} y=\int _{-\infty }^{\infty }x\cdot f_{Y}(x)\mathrm {d} x$

$E(gX)=\int _{-\infty }^{\infty }g(x)\cdot f(x)\mathrm {d} x$ $E(X^{2})=\int _{-\infty }^{\infty }x^{2}\cdot f(x)\mathrm {d} x$

$Var(X)=E((X-E(X))^{2}\$ $\sigma ={\sqrt {Var(X)}}$

$E((X-E(X))^{2})=E(X^{2}-2\cdot E(X)\cdot X+(E(X))^{2})=E(X^{2})-2\cdot E(X)\cdot E(X)+(E(X))^{2}=E(X^{2})-(E(X))^{2}$

$E(X^{2})=\sum _{k=1}^{\infty }k^{2}\cdot q^{k-1}\cdot p=...={\frac {2-p}{p^{2}}}$ $Var(X)=E(X^{2})-(E(X))^{2}={\frac {2-p}{p^{2}}}-{\frac {1}{p^{2}}}={\frac {1-p}{p^{2}}}={\frac {q}{p^{2}}}$

$Var(X)=E(X^{2})-(E(X))^{2}=...=p\cdot q$

$Var(X)=E(X^{2})-(E(X))^{2}={\frac {2-p}{p^{2}}}-{\frac {1}{p^{2}}}={\frac {1-p}{p^{2}}}={\frac {q}{p^{2}}}$

$E(X)={\frac {a+b}{2}}$

$E(X^{2})=\int _{-\infty }^{\infty }x^{2}\cdot f(x)\mathrm {d} x=...={\frac {b^{2}+a\cdot b+a^{2}}{3}}$ $Var(X)=E(X^{2})-(E(X))^{2}=...={\frac {(b-a)^{2}}{12}}$

$E(X)=(1-p)\cdot 0+p\cdot 1=p$

Mafi2

$\geq$ $\leq$

$(a_{n})_{n\in N}$ $\forall \epsilon <0\ \exists n_{0}\in N\ \forall n\geq n_{0}\ |a_{n}-a|<\epsilon$

$\lim _{n\to \infty }a_{n}=a=lim\;a_{n}$ $a_{n}{\overrightarrow {n\to \infty }}a\quad a_{n}\rightarrow a$

$f(x_{0})=\lim _{x\to x_{0}}f(x)$

$\forall \epsilon >0\quad \exists \delta >0\quad \forall x\quad |x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|<\delta$

${\frac {z}{w}}={\frac {xu+yv}{u^{2}+v^{2}}}+i*{\frac {yu-xv}{u^{2}+v^{2}}}$

$|z|={\sqrt {x^{2}+y^{2}}}={\sqrt {z\cdot {\overline {z}}}}$

$|z|={\sqrt {x^{2}+y^{2}}}$

$argz={\begin{cases}arccos{\frac {x}{|z|}},{\mbox{falls }}y\geq 0\\-arccos{\frac {x}{|z|}},{\mbox{falls }}y<0\end{cases}}$

$e^{i\cdot \zeta }=cos\zeta +i\cdot sin\zeta$

$e^{z}=e^{x+iy}=e^{x}\cdot e^{iy}=e^{x}(cosy+i\cdot siny)$

$|z|=e^{x}\quad$ $argz=y\pm 2k\pi \in (-\pi ,\pi ]$

1. $e^{i(\zeta +\psi )}=e^{i\zeta }\cdot e^{i\psi }$
2. $e^{i\cdot n\zeta }=(e^{i\zeta })^{n}$
3. ${\overline {e^{i\cdot \zeta }}}=e^{i(-\zeta )}={\frac {1}{e^{i\zeta }}}$

$z\in C{\mbox{ beliebig }}z=r\cdot e^{i\cdot \zeta }\quad (r=|z|{\mbox{ und }}\zeta =argz)$

${\sqrt[{k}]{z}}={\begin{Bmatrix}{\sqrt[{k}]{r}}\cdot e^{i\cdot {\frac {\zeta }{k}}},{\sqrt[{k}]{r}}\cdot e^{i\cdot {\frac {\zeta +2\pi }{k}}},{\sqrt[{k}]{r}}\cdot e^{i\cdot {\frac {\zeta +2\cdot 2\pi }{k}}},...,{\sqrt[{k}]{r}}\cdot e^{i\cdot {\frac {\zeta +(k-1)\cdot 2\pi }{k}}}\end{Bmatrix}}$

${\mbox{kgV}}(p(x),q(x))={\frac {p(x)\cdot q(x)}{{\mbox{ggT}}(p(x),q(x))}}$

$p(x)=\sum _{j=0}^{n}{y_{j}\cdot p_{j}(x)}$

$p_{j}(x)=\prod _{i\in {\begin{Bmatrix}0,...,n\end{Bmatrix}}\setminus {\begin{Bmatrix}j\end{Bmatrix}}}{\frac {x-x_{i}}{x_{j}-x_{i}}}$

$a_{n}=y_{0,n}\quad$

$(a_{n})_{n\in N}$ $(b_{n})_{n\in N}$ $(c_{n})_{n\in N}$

1. $\lim _{n\to \infty }{(a_{n}+b_{n})}=a+b$
2. $\lim _{n\to \infty }{(a_{n}\cdot b_{n})}=a\cdot b$
3. $\lim _{n\to \infty }{({\frac {a_{n}}{b_{n}}})}={\frac {a}{b}}{\mbox{ falls }}b\neq 0{\mbox{ und }}b_{n}\neq 0{\mbox{ fuer alle }}n\in N$
4. $\lim _{n\to \infty }{|a_{n}|}=|a|$
5. $\lim _{n\to \infty }{\sqrt {|a_{n}|}}={\sqrt {|a|}}$

$a_{n}\leq b_{n}\leq c_{n}$$n\geq k$ $\lim _{n\to \infty }{a_{n}}=\lim _{n\to \infty }{c_{n}}=c$ $\lim _{n\to \infty }{b_{n}}=c$