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Change result -> outcome

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Antti H S Laaksonen 2017-04-21 09:30:15 +03:00
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chapter24.tex
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@ -18,9 +18,9 @@ each outcome is $1/6$.
For example, we can calculate the following probabilities: For example, we can calculate the following probabilities:
\begin{itemize}[noitemsep] \begin{itemize}[noitemsep]
\item $P(\textrm{''the result is 4''})=1/6$ \item $P(\textrm{''the outcome is 4''})=1/6$
\item $P(\textrm{''the result is not 6''})=5/6$ \item $P(\textrm{''the outcome is not 6''})=5/6$
\item $P(\textrm{''the result is even''})=1/2$ \item $P(\textrm{''the outcome is even''})=1/2$
\end{itemize} \end{itemize}
\section{Calculation} \section{Calculation}
@ -79,7 +79,7 @@ where $X$ contains all possible outcomes
and $A$ is a subset of outcomes. and $A$ is a subset of outcomes.
For example, when drawing a dice, the outcomes are For example, when drawing a dice, the outcomes are
\[X = \{1,2,3,4,5,6\}.\] \[X = \{1,2,3,4,5,6\}.\]
Now, for example, the event ''the result is even'' Now, for example, the event ''the outcome is even''
corresponds to the set corresponds to the set
\[A = \{2,4,6\}.\] \[A = \{2,4,6\}.\]
@ -91,7 +91,7 @@ of probabilities of outcomes using the formula
For example, when throwing a dice, For example, when throwing a dice,
$p(x)=1/6$ for each outcome $x$, $p(x)=1/6$ for each outcome $x$,
so the probability of the event so the probability of the event
''the result is even'' is ''the outcome is even'' is
\[p(2)+p(4)+p(6)=1/2.\] \[p(2)+p(4)+p(6)=1/2.\]
The total probability of the outcomes in $X$ must The total probability of the outcomes in $X$ must
@ -144,11 +144,11 @@ is calculated using the formula
\[P(A \cup B)=P(A)+P(B)-P(A \cap B).\] \[P(A \cup B)=P(A)+P(B)-P(A \cap B).\]
For example, when throwing a dice, For example, when throwing a dice,
the union of the events the union of the events
\[A=\textrm{''the result is even''}\] \[A=\textrm{''the outcome is even''}\]
and and
\[B=\textrm{''the result is less than 4''}\] \[B=\textrm{''the outcome is less than 4''}\]
is is
\[A \cup B=\textrm{''the result is even or less than 4''},\] \[A \cup B=\textrm{''the outcome is even or less than 4''},\]
and its probability is and its probability is
\[P(A \cup B) = P(A)+P(B)-P(A \cap B)=1/2+1/2-1/6=5/6.\] \[P(A \cup B) = P(A)+P(B)-P(A \cap B)=1/2+1/2-1/6=5/6.\]
@ -174,8 +174,8 @@ Using the above sets,
\[P(A | B)= 1/3,\] \[P(A | B)= 1/3,\]
because the outcomes of $B$ are because the outcomes of $B$ are
$\{1,2,3\}$, and one of them is even. $\{1,2,3\}$, and one of them is even.
This is the probability of an even result This is the probability of an even outcome
if we know that the result is between $1 \ldots 3$. if we know that the outcome is between $1 \ldots 3$.
\subsubsection{Intersection} \subsubsection{Intersection}
@ -208,8 +208,8 @@ A \key{random variable} is a value that is generated
by a random process. by a random process.
For example, when throwing two dice, For example, when throwing two dice,
a possible random variable is a possible random variable is
\[X=\textrm{''the sum of the results''}.\] \[X=\textrm{''the sum of the outcomes''}.\]
For example, if the results are $[4,6]$ For example, if the outcomes are $[4,6]$
(meaning that we first throw a four and then a six), (meaning that we first throw a four and then a six),
then the value of $X$ is 10. then the value of $X$ is 10.
@ -232,7 +232,7 @@ The expected value can be calculated as the sum
where $x$ goes through all possible values of $X$. where $x$ goes through all possible values of $X$.
For example, when throwing a dice, For example, when throwing a dice,
the expected result is the expected outcome is
\[1/6 \cdot 1 + 1/6 \cdot 2 + 1/6 \cdot 3 + 1/6 \cdot 4 + 1/6 \cdot 5 + 1/6 \cdot 6 = 7/2.\] \[1/6 \cdot 1 + 1/6 \cdot 2 + 1/6 \cdot 3 + 1/6 \cdot 4 + 1/6 \cdot 5 + 1/6 \cdot 6 = 7/2.\]
A useful property of expected values is \key{linearity}. A useful property of expected values is \key{linearity}.