Proof of equation 51 when x and y are discrete we

Proof of Equation (5.1) when X and Y Are Discrete: We must show that
E [ X ] =
y
E [ X | Y = y ] P { Y = y } (5.2)

334 Chapter 7 Properties of Expectation
Now, the right-hand side of Equation (5.2) can be written as
y
E [ X | Y = y ] P { Y = y } =
y x
xP { X = x | Y = y } P { Y = y }
=
y x
x
P { X = x , Y = y }
P { Y = y }
P { Y = y }
=
y x
xP { X = x , Y = y }
=
x
x
y
P { X = x , Y = y }
=
x
xP { X = x }
= E [ X ]
and the result is proved.
One way to understand Equation (5.2) is to interpret it as follows: To calculate
E [ X ], we may take a weighted average of the conditional expected value of X given
that Y = y , each of the terms E [ X | Y = y ] being weighted by the probability of
the event on which it is conditioned. (Of what does this remind you?) This is an
extremely useful result that often enables us to compute expectations easily by first
conditioning on some appropriate random variable. The following examples illustrate
its use.
EXAMPLE 5c
A miner is trapped in a mine containing 3 doors. The first door leads to a tunnel that
will take him to safety after 3 hours of travel. The second door leads to a tunnel that
will return him to the mine after 5 hours of travel. The third door leads to a tunnel
that will return him to the mine after 7 hours. If we assume that the miner is at all
times equally likely to choose any one of the doors, what is the expected length of
time until he reaches safety?

Section 7.5 Conditional Expectation 335
E [ X | Y = 2] = 5 + E [ X ]. The argument behind the other equalities in Equation (5.3)
is similar. Hence,
E [ X ] =
1
3
( 3 + 5 + E [ X ] + 7 + E [ X ] )
or
E [ X ] = 15 .
EXAMPLE 5d Expectation of a sum of a random number of random variables
Suppose that the number of people entering a department store on a given day is
a random variable with mean 50. Suppose further that the amounts of money spent
by these customers are independent random variables having a common mean of $8.
Finally, suppose also that the amount of money spent by a customer is also inde-
pendent of the total number of customers who enter the store. What is the expected
amount of money spent in the store on a given day?