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Appendices

APPENDIX A. USEFUL TOOLS IN APPLIED PROBABILITY

This appendix summarizes some basic tools that can be found in most introductory
texts on probability.

Law of total expectation
In many applied probability problems it is only possible to compute certain prob-
abilities and expectations by using appropriate conditioning arguments. Since con-
ditional expectations are based on additional information, they are often easier to
compute than unconditional expectations. The law of total expectation states that,
for any two random variables X and Y defined on the same probability space,

E(X) = E(X | Y = y)P {Y = y} (A.1)
y
when Y has a discrete distribution and
∞

E(X) = E(X | Y = y)f (y) dy (A.2)
−∞
when Y has a continuous distribution with probability density f (y). It is assumed
that the relevant expectations exist. The law of total probability is a special case
of the law of total expectation:

P {X ≤ x} = P {X ≤ x | Y = y}P {Y = y} (A.3)
y
when Y has a discrete distribution and
∞

P {X ≤ x} = P {X ≤ x | Y = y}f (y) dy (A.4)
−∞
A First Course in Stochastic Models H.C. Tijms
c 2003 John Wiley & Sons, Ltd. ISBNs: 0-471-49880-7 (HB); 0-471-49881-5 (PB)