diff --git a/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex b/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex index 9ddfd18..56c5778 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex @@ -10,6 +10,9 @@ \begin{itemize} \item $\cX \sim \text{Ber}(p)$: $\E[\cX] = p$ ($\E[1_A] = \P[A]$) \item $\cX \sim \text{Bin}(n, p)$: $\E[\cX] = np$ + \item $\cX \sim \text{NBin}(r, p)$: $\E[\cX] = \frac{r(1 - p)}{p}$ + \item $\cX \sim \text{Geom}(p)$: $\E[\cX] = \frac{1}{p}$ + \item $\cX \sim \text{H}(n, r, m)$: $\E[\cX] = m\frac{n}{r}$ \item $\cX \sim \text{Poisson}(\lambda)$: $\E[\cX] = \lambda$ \end{itemize} diff --git a/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex index 5495d52..5989366 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex @@ -16,10 +16,14 @@ $\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$ \shortproposition $\cX_k$ paarw. unabh. $\V\left[ \sum_{k = 1}^{n} \cX_k \right] = \sum_{k = 1}^{n} \V[\cX_k]$. Falls $\cX_k$ abhängig, dann gilt $\neq$ +\newpage \shortexample Varianz von bekannten Verteilungen \begin{itemize} \item $\cX \sim \text{Ber}(p)$, $\V[\cX] = p (1 - p)$ \item $\cX \sim \text{Bin}(n, p)$, $\V[\cX] = n p (1 - p)$ + \item $\cX \sim \text{NBin}(r, p)$, $\V[\cX] = \frac{r(1 - p)}{p^2}$ + \item $\cX \sim \text{Geom}(p)$, $\V[\cX] = \frac{1 - p}{p^2}$ + \item $\cX \sim \text{H}(n, r, m)$, $\V[\cX] = m\frac{rn - r^2}{n^2} \frac{n - m}{n - 1}$ \item $\cX \sim \text{Poisson}(\lambda)$, $\V[\cX] = \lambda = \E[\cX]$ \item $\cX \sim \cU([a, b])$, $\V[\cX] = \frac{(b - a)^2}{12}$ \item $\cX \sim \cN(\mu, \sigma^2)$, $\V[\cX] = \sigma^2$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex b/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex index 03943d7..98fe343 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex @@ -1,4 +1,3 @@ -\newpage \subsection{Kovarianz} \shortdefinition $\cov(\cX, \cY) = \E[(\cX - \E[\cX])(\cY - \E[\cY])]$ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index ccae453..8daf9ef 100644 Binary files a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf and b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf differ