Fundamentals of Probability and Statistics for Engineers

which the value at Hence, is itself a random
variable, and one of its very useful properties is that

If Y is a discrete random variable taking on values y 1 ,y 2 ,..., theabovestates
that

and

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

Since, from Equation (2.24),

we have

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).

84 Fundamentals of Probability and Statistics for Engineers

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