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\title{Independence\footnote{ This slide show is an open-source document. See last slide for copyright information.}}
\subtitle{STA 256: Fall 2018}
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\begin{frame}
\frametitle{Independence: The idea} \pause
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\begin{itemize}
    \item Independent means totally unrelated.  \pause
    \item If we say that taking Vitamin C supplements in \emph{independent} of whether you get cancer\pause, it means that taking Vitamin C supplements has \emph{no connection} to whether you get cancer or not. \pause
    \item It's a strong statement. \pause
    \item It has a precise technical definition.
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\frametitle{Definition of independence}
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Suppose $P(B)>0$, and $ P(A|B) = P(A)$. \pause
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\begin{eqnarray*}
  &  &  \frac{P(A \cap B)}{P(B)} = P(A) \\ \pause
  & \Rightarrow \pause &  P(A \cap B)  = P(A) \, P(B) \pause
\end{eqnarray*} 
We use this \emph{definition}. \pause We say the events $A$ and $B$ are independent when \pause
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    P(A \cap B)  = P(A) \, P(B)
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It's symmetric, and applies even if $P(A)=0$ or $P(B)=0$.
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\frametitle{Definition of \emph{mutual} independence}
\framesubtitle{Of several events} \pause

A set of events $A_1, \ldots, A_n$ are \emph{mutually independent} if the probability of the intersection of any sub-collection is the product of probabilities. \vspace{5mm}  \pause 

Pairwise is not enough. \pause Example from the text, P. 24: \pause \vspace{2mm}

A fair coin is tossed twice. \pause Outcomes are \hspace{3mm} HH, HT, TH, TT  \pause \linebreak
Let \pause
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    \item[]$A$ = Head on first toss. \pause 
    \item[]$B$ = Head on second toss. \pause 
    \item[]$C$ = Exactly one Head. \pause 
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$P(A)=P(B)=P(C) = \frac{1}{2}$ \pause and they are pairwise independent\pause, but \pause $P(A \cap B \cap C) = \pause P(\emptyset) = 0 \pause \neq \frac{1}{8}$.

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\frametitle{The dice have no memory} \pause 
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Successive outcomes of simple statistical experiments like flipping a coin or rolling a die will always be assumed mutually independent.
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\frametitle{Copyright Information}

This slide show was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:

\vspace{5mm}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}

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$ = \{\omega \in \Omega: \}$

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