Notes in Probability Theory

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Published	11/14/2024	[latex]Definition \textbf{bedingte Wahrscheinlichkeit} \( P( B \| A ) = \ldots \)[/latex]
Published	11/14/2024	[latex]Erwartungswert einer diskreten Zufallsvariable \( X \)[/latex]
Published	11/14/2024	[latex](Wahrscheinlichkeitstheorie) Fallunterscheidungsformel[/latex]
Published	11/14/2024	[latex](Wahrscheinlichkeitstheorie) Formel von Bayes[/latex]
Published	11/14/2024	[latex]Wann gilt \( \mathbb{E} [ XY ] = \mathbb{E} [ X ] \mathbb{E} [ Y ] \)?[/latex]
Published	11/14/2024	[latex]Definition \textbf{Erwartungswert} einer reellen Zufallsvariablen (allgemein)[/latex]
Published	11/14/2024	[latex]Central Limit Theorem[/latex]
Published	11/14/2024	[latex]Jensen's inequality[/latex]
Published	11/14/2024	[latex]Probability integral transform[/latex]
Published	11/14/2024	[latex]The \textbf{covariance} of two random variables \( X \) and \( Y \) is ...[/latex]
Published	11/14/2024	[latex]Write the covariance as a sum of multiple expected values[/latex]
Published	11/14/2024	[latex]Express the variance through a single expectation [/latex]
Published	11/14/2024	[latex]Express the \textbf{variance} through 2 expectations [/latex]
Published	11/14/2024	[latex]A \textbf{random variable} is ... [/latex]
Published	11/14/2024	[latex]Let \( X\colon (\Omega, \mathcal{A}) \to (\mathbb{R}, \mathcal{\mathbb{R}}) \) be a random variable. For \( M \in \mathcal{B}( \mathbb{R} ) \)…
Published	11/14/2024	[latex]A family of events \( (A_i)_{i \in I} \) is called \textbf{independent} if ... [/latex]
Published	11/14/2024	[latex]A family \( (X_i)_{i \in I} \) of random varaibles is called \textbf{independent} if ... [/latex]
Published	11/14/2024	[latex]The \textbf{covariance} of two random variables \( X,Y \) is ... [/latex]
Published	11/14/2024	[latex]Show: Independent random variables are uncorrelated [/latex]
Published	11/14/2024	[latex]\begin{align} \Var( X ) = 0 \iff \ldots \end{align} [/latex]
Published	11/14/2024	[latex]Multivariate Central Limit Theorem [/latex]
Published	11/14/2024	[latex]Are uncorrelated random variables independent? If yes: explain why. If not: are there special cases? [/latex]
Published	11/14/2024	[latex]Let \( p \) be a probability distribution on a countable set \( E \) Definition: \textbf{Entropy} of \( p \) [/latex]
Published	11/14/2024	[latex]A sequence of random variables \( (X_n)_{n \in \mathbb{N}} \) in a metrix space \( E \) \textbf{converges in distribution} to \( X \) (\( X_n \…
Published	11/14/2024	[latex]Slutzky's theorem [/latex]
Published	11/14/2024	[latex]Markov inequality[/latex]
Published	11/14/2024	[latex]Properties of the map \( \Cov\colon \mathcal{L}^2( \mathbb{P} ) \times \mathcal{L}^2( \mathbb{P} ) \to \mathbb{R} \) [/latex]
Published	11/14/2024	[latex]Cauchy-Schwarz inequality (with Covariance) [/latex]
Published	11/14/2024	[latex]Bienayme formula [/latex]
Published	11/14/2024	[latex]What can we say about \( \Cov( X, c ) \) if \( c \) is a constant? [/latex]
Published	11/14/2024	[latex]Weak law of large numbers [/latex]
Status	Last Update	Fields