* Probability
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:ANKI_DECK: math::probability
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** What is the Gamma distribution a generalisation of?
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the *exponential* distribution


** Which 2 distributions are memoryless? What does this mean?
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1. Geometric (discrete)
2. Exponential (continuous)

\[\mathbb{P}(T> t_1+t_2 | T > t_1) = \mathbb{P}(T>t_2)\]


** Law of Total Probability
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Suppose that $\{A_i\}^k_{i=1}$ forms a partition of the sample space $\Omega$. Then for any event $B$, $\mathcal{P}(B)=\ldots$

*** Back

\begin{align}
\mathbb{P}(B) &= \sum_{i=1}^k \mathbb{P}(B\cap A_i) \\
&= \sum_{i=1}^k \mathbb{P}(B|A_i)\mathbb{P}(A_i)
\end{align}

\begin{tikzpicture}
    % Sample Space
    \draw (0,0) rectangle (6,4) node[below left] {$\Omega$};
    
    % Partition lines
    \draw (2,0) -- (2,4);
    \draw (4,0) -- (4,4);
    
    % Labels for partition
    \node at (1,3.5) {$A_1$};
    \node at (3,3.5) {$A_2$};
    \node at (5,3.5) {$A_3$};
    
    % Event B (Ellipse)
    \draw[thick, fill=blue!20, opacity=0.5] (3,2) ellipse (2cm and 1cm);
    \node at (3,2) {$B$};
    
    % Intersections (Optional labels)
    \node[scale=0.7] at (1.5,2) {$B \cap A_1$};
    \node[scale=0.7] at (3,2.5) {$B \cap A_2$};
    \node[scale=0.7] at (4.5,2) {$B \cap A_3$};
\end{tikzpicture}


** Is the Geometric distribution a *discrete* or *continuous* distribution? What is it a characterisation of?
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Discrete.

Characterisation of "the number of trials required to see the first Bernoulli success"


** What is the PMF of the Geometric distribution?
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\[p_X(x) = \mathbb{P}(X=x) = \underbrace{(1-\pi)^{x-1}}_{\text{seeing $x-1$ failures}} \times \overbrace{\pi}^{\text{seeing success on the $x^\textup{th}$ trial}}\]


** What are the expansions of the following sets? What is the corresponding law called?
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\[(A\cup B)\cap C = \]

\[(A\cap B)\cup C = \]


*** Back

"Distributive Law"

\[(A\cup B)\cap C = (A\cap C)\cup (B\cap C) \]

\[(A\cap B)\cup C = (A\cup C)\cap (B \cup C) \]

** The /conditional probability/ that an event $A$ occurs given that an event $B$ has occurred is $\mathbb{P}(A|B)=$
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\[\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)},\qquad B\neq \varnothing\]


** Two events $A$ and $B$ are independent if and only if?
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\[\mathbb{P}(A\cap B) = \mathbb{P}(A)\times \mathbb{P}(B)\]

note: you can derive this from conditional probability.


** What does the *associative law* for sets look like?
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\[(A\cup B)\cup C = A\cup (B \cup C) \]

\[(A\cap B)\cap C = A\cap (B \cap C) \]

** Gamma function
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\[\Gamma(\alpha) = ?\]

*** Back

\[\Gamma(\alpha) = \int_0^\infty t^{\alpha-1} e^{-t}\; \mathrm{d}t\]