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Lienare Algebra & Vektoren
P r : Ω → [ 0 , 1 ] {\displaystyle Pr:\Omega \rightarrow [0,1]}
∑ a ∈ A P r ( a ) = 1 {\displaystyle \sum _{a\in A}Pr(a)=1}
A ⊆ Ω {\displaystyle A\subseteq \Omega }
P r : P ( Ω ) → [ 0 , 1 ] {\displaystyle Pr:{\mathcal {P}}(\Omega )\rightarrow [0,1]}
P r ( A ) = ∑ a ∈ A P r ( a ) {\displaystyle Pr(A)=\sum _{a\in A}Pr(a)}
P ( Ω ) = 2 Ω {\displaystyle P(\Omega )=2^{\Omega }}
1 6 {\displaystyle {\frac {1}{6}}}
¬ {\displaystyle \neg }
∪ {\displaystyle \cup } ∈ {\displaystyle \in }
∩ {\displaystyle \cap }
⊆ {\displaystyle \subseteq }
∅ {\displaystyle \varnothing }
P r ( B ) = ( 5 3 ) ⋅ 1 2 5 {\displaystyle Pr(B)={5 \choose 3}\cdot {\frac {1}{2^{5}}}}
P r ( A ) = 5 6 k − 1 ⋅ 1 6 = 5 k − 1 6 k {\displaystyle Pr(A)={5 \over 6}^{k-1}\cdot {1 \over 6}={5^{k-1} \over 6^{k}}}
A ∈ F ⇒ A ¯ ∈ F {\displaystyle A\in {\mathcal {F}}\Rightarrow {\overline {A}}\in {\mathcal {F}}}
A 1 , A 2 , . . . ∈ F ⇒ ⋃ i = 1 ∞ A i ∈ F {\displaystyle A_{1},A_{2},...\in {\mathcal {F}}\Rightarrow \bigcup _{i=1}^{\infty }A_{i}\in {\mathcal {F}}}
A ¯ {\displaystyle {\overline {A}}}
A i ∈ F {\displaystyle A_{i}\in {\mathcal {F}}}
P r ( ⋃ i = 1 ∞ A i ) = ∑ i = 1 ∞ P r ( A i ) {\displaystyle Pr\left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }Pr(A_{i})}
F ⊆ P ( Ω ) {\displaystyle {\mathcal {F}}\subseteq P(\Omega )}
P r ( A | B ) = P r ( A ∩ B ) P r ( B ) {\displaystyle Pr(A|B)={\frac {Pr(A\cap B)}{Pr(B)}}}
P r ( A ) = ∑ i P r ( A | B i ) ⋅ P r ( B i ) {\displaystyle Pr(A)=\sum _{i}Pr(A|B_{i})\cdot Pr(B_{i})}
E ( X ) = ∑ w ∈ Ω X ( w ) ⋅ P r ( w ) {\displaystyle E(X)=\sum _{w\in \Omega }X(w)\cdot Pr({w})}
I m X = { x ∈ R | ∃ a ∈ Ω X ( a ) = x } {\displaystyle ImX=\{x\in R|\exists a\in \Omega \quad X(a)=x\}}
∀ x ∈ R X − 1 ( x ) = { a ∈ Ω | x ( a ) = x } ∈ F {\displaystyle \forall x\in R\quad X^{-1}(x)=\{a\in \Omega |x(a)=x\}\in {\mathcal {F}}}
E ( X ) = ∑ x ∈ I m ( X ) x ⋅ P r ( X − 1 ( x ) ) {\displaystyle E(X)=\sum _{x\in Im(X)}x\cdot Pr(X^{-1}(x))}
X − 1 ( T ) ∈ F {\displaystyle X^{-1}(T)\in {\mathcal {F}}}
X − 1 ( T ) = ⋃ x ∈ ( T ∩ I m X ) X − 1 ( T ) {\displaystyle X^{-1}(T)=\bigcup _{x\in (T\cap ImX)}X^{-1}(T)}
P r ( X − 1 ( x ) ) = P r X ( x ) = P r ( X = x ) {\displaystyle Pr(X^{-1}(x))=Pr_{X}(x)=Pr(X=x)\ }
P r X ( k ) = ( n k ) ⋅ p k ⋅ q n − k {\displaystyle Pr_{X}(k)={n \choose k}\cdot p^{k}\cdot q^{n-k}} k ∈ { 0 , 1 , . . . , n } {\displaystyle k\in \{0,1,...,n\}}
P r X ( k ) = p ⋅ q k − 1 {\displaystyle Pr_{X}(k)=p\cdot q^{k-1}} k ∈ N + {\displaystyle k\in \mathbb {N} ^{+}}
N + {\displaystyle \mathbb {N} ^{+}}
1 6 ( 5 6 ) n − 1 {\displaystyle {\frac {1}{6}}\left({\frac {5}{6}}\right)^{n-1}}
P r X ( k ) = 1 k ! λ k e − λ {\displaystyle Pr_{X}(k)={\frac {1}{k!}}\lambda ^{k}e^{-\lambda }}
{ a ∈ Ω | X ( a ) ≤ x } ∈ F {\displaystyle \{a\in \Omega |X(a)\leq x\}\in {\mathcal {F}}}
F X ( x ) = P r ( { a ∈ Ω | X ( a ) ≤ x } ) = P r ( X ≤ x ) {\displaystyle F_{X}(x)=Pr(\{a\in \Omega |X(a)\leq x\})=Pr(X\leq x)}
F X ( x ) = ∑ y ≤ x P r X ( y ) {\displaystyle F_{X}(x)=\sum _{y\leq x}Pr_{X}(y)}
x ≤ y ⇒ F ( x ) ≤ F ( y ) {\displaystyle x\leq y\Rightarrow F(x)\leq F(y)}
lim x → − ∞ F ( x ) = 0 {\displaystyle \lim _{x\to -\infty }F(x)=0} lim x → ∞ F ( x ) = 1 {\displaystyle \lim _{x\to \infty }F(x)=1}
∀ x ∈ R lim h → 0 + F ( x + h ) = F ( x ) {\displaystyle \forall x\in \mathbb {R} \lim _{h\to 0+}F(x+h)=F(x)}
F X ( x ) = ∫ − ∞ x f ( t ) d t {\displaystyle F_{X}(x)=\int _{-\infty }^{x}f(t)\mathrm {d} t}
E ( X ) = ∑ x ∈ I m X x ⋅ P r X ( x ) = ∑ x ∈ I m x x ⋅ P r ( { a ∈ Ω | X ( a ) = x } ) {\displaystyle 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 ) = ∫ − ∞ ∞ x ⋅ f ( x ) d x {\displaystyle E(X)=\int _{-\infty }^{\infty }x\cdot f(x)\mathrm {d} x}
X = X 1 + X 2 + X 3 + . . . + X n {\displaystyle X=X_{1}+X_{2}+X_{3}+...+X_{n}\ }
E ( X ) = E ( X 1 ) + E ( X 2 ) + E ( X 3 ) + . . . + E ( X n ) = n ⋅ p {\displaystyle E(X)=E(X_{1})+E(X_{2})+E(X_{3})+...+E(X_{n})=n\cdot p}
P r X ( k ) = ( e − p ) k − 1 ⋅ p = q k − 1 ⋅ p {\displaystyle Pr_{X}(k)=(e-p)^{k-1}\cdot p=q^{k-1}\cdot p}
E ( X ) = ∑ k = 1 ∞ k ⋅ q k − 1 ⋅ p = . . . = 1 ⋅ 1 1 − q = 1 p {\displaystyle E(X)=\sum _{k=1}^{\infty }k\cdot q^{k-1}\cdot p=...=1\cdot {\frac {1}{1-q}}={\frac {1}{p}}}
E ( X ) = ∑ k = 0 ∞ k ⋅ 1 k ! ⋅ λ k ⋅ e − λ = . . . = λ ⋅ e λ ⋅ e − λ = λ {\displaystyle 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 }
1 b − a {\displaystyle {\frac {1}{b-a}}}
E ( X ) = ∫ − ∞ ∞ x ⋅ f ( x ) d x = ∫ a b x ⋅ 1 b − a d x = . . . = a + b 2 {\displaystyle 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 : Ω → R ≥ 0 {\displaystyle X:\Omega \rightarrow \mathbb {R} ^{\geq 0}}
P r ( X ≥ t ) ≤ E ( X ) t {\displaystyle Pr(X\geq t)\leq {\frac {E(X)}{t}}} P r ( X ≥ α E ( X ) ) ≤ 1 α {\displaystyle Pr(X\geq \alpha E(X))\leq {\frac {1}{\alpha }}}
t = 2 , 0 m P r ( X ≥ 2 , 0 ) ≤ E ( X ) t = 1 , 7 2 , 0 = 0 , 85 {\displaystyle t=2{,}0m\quad Pr(X\geq 2{,}0)\leq {\frac {E(X)}{t}}={\frac {1{,}7}{2{,}0}}=0{,}85}
t = 0 , 5 m P r ( Y ≥ 0 , 5 ) ≤ 0 , 2 0 , 5 = 0 , 4 {\displaystyle t=0{,}5m\quad Pr(Y\geq 0{,}5)\leq {\frac {0{,}2}{0{,}5}}=0{,}4}
E ( X 2 ) = ∑ x ∈ I m X x 2 ⋅ P r ( X = x ) {\displaystyle E(X^{2})=\sum _{x\in ImX}x^{2}\cdot Pr(X=x)}
E ( X ) = ∫ − ∞ ∞ x ⋅ f X ( x ) d x {\displaystyle E(X)=\int _{-\infty }^{\infty }x\cdot f_{X}(x)\mathrm {d} x}
E ( Y ) = ∫ − ∞ ∞ y ⋅ f Y ( y ) d y = ∫ − ∞ ∞ x ⋅ f Y ( x ) d x {\displaystyle 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 ( g X ) = ∫ − ∞ ∞ g ( x ) ⋅ f ( x ) d x {\displaystyle E(gX)=\int _{-\infty }^{\infty }g(x)\cdot f(x)\mathrm {d} x} E ( X 2 ) = ∫ − ∞ ∞ x 2 ⋅ f ( x ) d x {\displaystyle E(X^{2})=\int _{-\infty }^{\infty }x^{2}\cdot f(x)\mathrm {d} x}
V a r ( X ) = E ( ( X − E ( X ) ) 2 {\displaystyle Var(X)=E((X-E(X))^{2}\ } σ = V a r ( X ) {\displaystyle \sigma ={\sqrt {Var(X)}}}
E ( ( X − E ( X ) ) 2 ) = E ( X 2 − 2 ⋅ E ( X ) ⋅ X + ( E ( X ) ) 2 ) = E ( X 2 ) − 2 ⋅ E ( X ) ⋅ E ( X ) + ( E ( X ) ) 2 = E ( X 2 ) − ( E ( X ) ) 2 {\displaystyle 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 ) = ∑ k = 1 ∞ k 2 ⋅ q k − 1 ⋅ p = . . . = 2 − p p 2 {\displaystyle E(X^{2})=\sum _{k=1}^{\infty }k^{2}\cdot q^{k-1}\cdot p=...={\frac {2-p}{p^{2}}}} V a r ( X ) = E ( X 2 ) − ( E ( X ) ) 2 = 2 − p p 2 − 1 p 2 = 1 − p p 2 = q p 2 {\displaystyle 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}}}}
V a r ( X ) = E ( X 2 ) − ( E ( X ) ) 2 = . . . = p ⋅ q {\displaystyle Var(X)=E(X^{2})-(E(X))^{2}=...=p\cdot q}
V a r ( X ) = E ( X 2 ) − ( E ( X ) ) 2 = 2 − p p 2 − 1 p 2 = 1 − p p 2 = q p 2 {\displaystyle 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 ) = a + b 2 {\displaystyle E(X)={\frac {a+b}{2}}}
E ( X 2 ) = ∫ − ∞ ∞ x 2 ⋅ f ( x ) d x = . . . = b 2 + a ⋅ b + a 2 3 {\displaystyle E(X^{2})=\int _{-\infty }^{\infty }x^{2}\cdot f(x)\mathrm {d} x=...={\frac {b^{2}+a\cdot b+a^{2}}{3}}} V a r ( X ) = E ( X 2 ) − ( E ( X ) ) 2 = . . . = ( b − a ) 2 12 {\displaystyle Var(X)=E(X^{2})-(E(X))^{2}=...={\frac {(b-a)^{2}}{12}}}
E ( X ) = ( 1 − p ) ⋅ 0 + p ⋅ 1 = p {\displaystyle E(X)=(1-p)\cdot 0+p\cdot 1=p}
≥ {\displaystyle \geq } ≤ {\displaystyle \leq }
( a n ) n ∈ N {\displaystyle (a_{n})_{n\in N}} ∀ ϵ < 0 ∃ n 0 ∈ N ∀ n ≥ n 0 | a n − a | < ϵ {\displaystyle \forall \epsilon <0\ \exists n_{0}\in N\ \forall n\geq n_{0}\ |a_{n}-a|<\epsilon }
lim n → ∞ a n = a = l i m a n {\displaystyle \lim _{n\to \infty }a_{n}=a=lim\;a_{n}} a n n → ∞ → a a n → a {\displaystyle a_{n}{\overrightarrow {n\to \infty }}a\quad a_{n}\rightarrow a}
f ( x 0 ) = lim x → x 0 f ( x ) {\displaystyle f(x_{0})=\lim _{x\to x_{0}}f(x)}
∀ ϵ > 0 ∃ δ > 0 ∀ x | x − x 0 | < δ ⇒ | f ( x ) − f ( x 0 ) | < δ {\displaystyle \forall \epsilon >0\quad \exists \delta >0\quad \forall x\quad |x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|<\delta }
z w = x u + y v u 2 + v 2 + i ∗ y u − x v u 2 + v 2 {\displaystyle {\frac {z}{w}}={\frac {xu+yv}{u^{2}+v^{2}}}+i*{\frac {yu-xv}{u^{2}+v^{2}}}}
| z | = x 2 + y 2 = z ⋅ z ¯ {\displaystyle |z|={\sqrt {x^{2}+y^{2}}}={\sqrt {z\cdot {\overline {z}}}}}
| z | = x 2 + y 2 {\displaystyle |z|={\sqrt {x^{2}+y^{2}}}}
a r g z = { a r c c o s x | z | , falls y ≥ 0 − a r c c o s x | z | , falls y < 0 {\displaystyle argz={\begin{cases}arccos{\frac {x}{|z|}},{\mbox{falls }}y\geq 0\\-arccos{\frac {x}{|z|}},{\mbox{falls }}y<0\end{cases}}}
e i ⋅ ζ = c o s ζ + i ⋅ s i n ζ {\displaystyle e^{i\cdot \zeta }=cos\zeta +i\cdot sin\zeta }
e z = e x + i y = e x ⋅ e i y = e x ( c o s y + i ⋅ s i n y ) {\displaystyle e^{z}=e^{x+iy}=e^{x}\cdot e^{iy}=e^{x}(cosy+i\cdot siny)}
| z | = e x {\displaystyle |z|=e^{x}\quad } a r g z = y ± 2 k π ∈ ( − π , π ] {\displaystyle argz=y\pm 2k\pi \in (-\pi ,\pi ]}
z ∈ C beliebig z = r ⋅ e i ⋅ ζ ( r = | z | und ζ = a r g z ) {\displaystyle z\in C{\mbox{ beliebig }}z=r\cdot e^{i\cdot \zeta }\quad (r=|z|{\mbox{ und }}\zeta =argz)}
z k = { r k ⋅ e i ⋅ ζ k , r k ⋅ e i ⋅ ζ + 2 π k , r k ⋅ e i ⋅ ζ + 2 ⋅ 2 π k , . . . , r k ⋅ e i ⋅ ζ + ( k − 1 ) ⋅ 2 π k } {\displaystyle {\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}}}
kgV ( p ( x ) , q ( x ) ) = p ( x ) ⋅ q ( x ) ggT ( p ( x ) , q ( x ) ) {\displaystyle {\mbox{kgV}}(p(x),q(x))={\frac {p(x)\cdot q(x)}{{\mbox{ggT}}(p(x),q(x))}}}
p ( x ) = ∑ j = 0 n y j ⋅ p j ( x ) {\displaystyle p(x)=\sum _{j=0}^{n}{y_{j}\cdot p_{j}(x)}}
p j ( x ) = ∏ i ∈ { 0 , . . . , n } ∖ { j } x − x i x j − x i {\displaystyle 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 {\displaystyle a_{n}=y_{0,n}\quad }
( a n ) n ∈ N {\displaystyle (a_{n})_{n\in N}} ( b n ) n ∈ N {\displaystyle (b_{n})_{n\in N}} ( c n ) n ∈ N {\displaystyle (c_{n})_{n\in N}}
a n ≤ b n ≤ c n {\displaystyle a_{n}\leq b_{n}\leq c_{n}} n ≥ k {\displaystyle n\geq k} lim n → ∞ a n = lim n → ∞ c n = c {\displaystyle \lim _{n\to \infty }{a_{n}}=\lim _{n\to \infty }{c_{n}}=c} lim n → ∞ b n = c {\displaystyle \lim _{n\to \infty }{b_{n}}=c}
![{\displaystyle Pr:\Omega \rightarrow [0,1]}](/media/api/rest_v1/media/math/render/svg/b3d9b18f3cc4102ab783966a78296e6baa4f11b9)
![{\displaystyle Pr:{\mathcal {P}}(\Omega )\rightarrow [0,1]}](/media/api/rest_v1/media/math/render/svg/1c85dbd1d9986a91323c2f6af08eefc963a640b4)
![{\displaystyle argz=y\pm 2k\pi \in (-\pi ,\pi ]}](/media/api/rest_v1/media/math/render/svg/52f954303f7bbedd1ab3cf07bdbfd306fc9436f9)
![{\displaystyle {\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}}}](/media/api/rest_v1/media/math/render/svg/6ddf79c54a74990d9782efdc6afd237c4e43c62d)
