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.

We are given joint pmf of \({\rm{X}}\) and\({\rm{Y}}\).

(a):

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 \({\rm{f}}\left( {{\rm{x, y}}} \right)\) pdf.

In our case, \({\rm{g(X,Y) = X + Y}}\), and the random variables are discrete, therefore the following is true

\(\begin{aligned}{\rm{E(X + Y) = }}\sum\limits_{\rm{x}} {\sum\limits_{\rm{y}} {\rm{g}} } {\rm{(x,y) \times p(x,y)}}\\{\rm{ = (0 + 0) \times 0}}{\rm{.02 + (0 + 5) \times 0}}{\rm{.06 + \ldots + (10 + 10) \times 0}}{\rm{.14 + (10 + 15) \times 0}}{\rm{.01}}\\{\rm{ = 14}}{\rm{.1}}\end{aligned}\)

(b):

We are interested in expectation of random variable\({\rm{g(X,Y) = max(X,Y)}}\). Therefore,

\(\begin{aligned}{\rm{E(max(X,Y)) = }}\sum\limits_{\rm{x}} {\sum\limits_{\rm{y}} {\rm{g}} } {\rm{(x,y) \times p(x,y)}}\\{\rm{ = max(0,0) \times 0}}{\rm{.02 + max(0,5) \times 0}}{\rm{.06 + \ldots + max(10,10) \times 0}}{\rm{.14 + max(10,15) \times 0}}{\rm{.01}}\\{\rm{ = 0 \times 0}}{\rm{.02 + 5 \times 0}}{\rm{.06 + \ldots + \ldots 10 \times 0}}{\rm{.14 + 15 \times 0}}{\rm{.01}}\\{\rm{ = 9}}{\rm{.6}}\end{aligned}\)