Section 5.2 Random Variables
For a given set of events, we might have difficulty doing mathematics since the outcomes are not numerical. In order to accomodate our desire to convert to numerical measures we want to assign numerical values to all outcomes. The process of doing this creates what is known as a random variable.
We will make various restrictions on the range of the random variable to fit different generalized problems. Then, we will be able to work on a problem (which may be inherently non-numerical) by using the random variable in subsequent calculations.
Example 5.2.2 Success vs Failure
When dealing with only two outcomes, one might use
\begin{equation*}
S = \text{{ success, failure }}.
\end{equation*}
Choose
\begin{gather*}
X(success)=1\\
X(failure)=0.
\end{gather*}
Then, R={0,1}.
Example 5.2.3 Standard Dice Pairs
When gambling with a pair of dice, one might use S=ordered pairs of all possible rolls. Then
\begin{equation*}
S = \text{ {(a,b): a=die 1 outcome, b=die 2 outcome}}.
\end{equation*}
Choose
\begin{equation*}
X( (a,b) ) = a+b.
\end{equation*}
Then, R={2, 3, 4, 5, ..., 12}.
Example 5.2.4 Other Dice Options
When rolling dice in a board game (like RISK), one might use
\begin{equation*}
S= \text{{(a,b): a=die 1 outcome, b=die 2 outcome}}
\end{equation*}
Choose
\begin{equation*}
X( (a,b) ) = \text{max{a,b}}.
\end{equation*}
Then, R={1, 2, 3, 4, 5, 6}.