Page 101 - Fundamentals of Probability and Statistics for Engineers

84 Fundamentals of Probability and Statistics for Engineers

which the value at Y ˆ y i is EfXjY ˆ y i g. Hence, EfXjYg is itself a random
variable, and one of its very useful properties is that

EfXgˆ EfEfXjYgg 4:13†

If Y is a discrete random variable taking on values y 1 , y 2 ,..., the above states
that
X
EfXgˆ EfXjY ˆ y i gP Y ˆ y i †; 4:14†
i
and
1
Z
EfXgˆ EfXjygf y†dy; 4:15†
Y
1
if Y is continuous.
To establish the relation given by Equation (4.13), let us show that Equation
(4.14) is true when both X and Y are discrete. Starting from the right-hand side
of Equation (4.14), we have

X X X
EfXjY ˆ y i gP Y ˆ y i †ˆ x j P X ˆ x j jY ˆ y i †P Y ˆ y i †:
i i j
Since, from Equation (2.24),

P X ˆ x j \ Y ˆ y i †
P X ˆ x j jY ˆ y i †ˆ ;
P Y ˆ y i †
we have

X X X
EfXjY ˆ y i gP Y ˆ y i †ˆ x j p XY x j ; y i †
i i j
X X
ˆ p x j ; y i †
x j
XY
j i
X
ˆ x j p x j †
X
j
ˆ EfXg;
and the desired result is obtained.
The usefulness of Equation (4.13) is analogous to what we found in using the
theorem of total probability discussed in Section 2.4 (see Theorem 2.1, page 23).

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