91Ó°ÊÓ

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes—how likely they are—when we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

The

Expected Value

(mean value) of a random variable\({\rm{g(X,Y)}}\), where \({\rm{g( \times )}}\) is a function, denoted as \({\rm{E(g(X,Y))}}\)is given by

\({\rm{E(g(X,Y)) = }}\left\{ {\begin{aligned}{*{20}{l}}{\sum\limits_{\rm{x}} {\sum\limits_{\rm{y}} {\rm{g}} } {\rm{(x,y) \times p(x,y)}}}&{{\rm{,X and Y discrete }}}\\{\int_{{\rm{ - ¥}}}^{\rm{¥}} {\int_{{\rm{ - ¥}}}^{\rm{¥}} {\rm{g}} } {\rm{(x,y) \times f(x,y)dxdy}}}&{{\rm{,X and Y continuous}}{\rm{. }}}\end{aligned}} \right.\)

where \({\rm{p(x,y)}}\) is pmf and \({\rm{f(x,y)}}\) pdf.

We want to use this. In order to do that, we need to determine \({\rm{g(X,Y)}}\)which would be the number of individuals (including A and B) who handle the message.

We first need to notice that individuals \({\rm{A}}\) and \({\rm{B}}\)cannot sit on the same spot. The minimum number of individuals who can handle the message is two (only A and B), and that is if they sit one to another. The maximum number of people who can handle the message is \({\rm{4}}\) (if there are two people sitting from the left and right of \({\rm{A}}\) and \({\rm{B}}\) - they sit across).

The sits are numbered, \({\rm{X}}\)represent A's seat number, and \({\rm{Y}}\) represents B's seat number, therefore, \({\rm{(2,1)}}\)would mean that person A sits on seat\({\rm{2}}\), and person B sits on seat\({\rm{1}}\), and \({\rm{g(2,1)}}\) would be \({\rm{2}}\) , because the only two individuals who handles the message are \({\rm{A}}\) and\({\rm{B}}\). We will do the same thing and represent it in a table.

To compute the expectation, we need\({\rm{p(x,y)}}\). There are

\({\rm{6 \times 5 = 30}}\)

total number of ways that \({\rm{A}}\) and \({\rm{B}}\) can sit, therefore

\({\rm{p(x,y) = }}\frac{{\rm{1}}}{{{\rm{30}}}}{\rm{.}}\)

The following is true

\(\begin{aligned}{\rm{E(g(X,Y)) = }}\sum\limits_{\rm{x}} {\sum\limits_{\rm{y}} {\rm{g}} } {\rm{(x,y) \times p(x,y)}}\\{\rm{ = }}\frac{{\rm{1}}}{{{\rm{30}}}}{\rm{ \times }}\sum\limits_{\rm{x}} {\sum\limits_{\rm{y}} {\rm{g}} } {\rm{(x,y)}}\\{\rm{ = }}\frac{{\rm{1}}}{{{\rm{30}}}}{\rm{ \times 84}}\\{\rm{ = 2}}{\rm{.8}}\end{aligned}\)