Part 4: Expected Value

Since a probability density is similar in concept to a mass density, we define the expected values of the random variables X and Y to be the coordinates of the center of mass of a lamina of the sample space S with density p( x,y) . However, since

p( x,y) dA = 1

this implies that the expected values of X and Y are given by

X  =  xp( x,y) dA,    Y  =  yp(x,y) dA

Often we denote the expected values as E( X) and E(Y) , respectively, and we assume they represent the most likely outcome of the experiment.      

EXAMPLE 7    What is the expected time for ``waiting for a table'' and ``completing the meal'' in example 6.       

Solution: Since the sample space is the 1st quadrant, the expected time for waiting on a table is

E( X) = x  1 300 e-x/10e-y/30dA =  1 300 ó õ ¥ 0  ó õ ¥ 0  xe-x/10e-y/30dydx =  1 10 ó õ ¥ 0  xe-x/10dx =  1 10 ·100 = 10

Likewise, E( Y) = 30. That is, the expected values of the 2 events are the average waiting time and the average dining time, respectively.