Page 88 - Probability and Statistical Inference

2

Expectations of Functions of

Random Variables

2.1 Introduction

In Chapter 1 we introduced the notions of discrete and continuous random
variables. We start Chapter 2 by discussing the concept of the expected value
of a random variable. The expected value of a random variable is sometimes
judged as the “center” of the probability distribution of the variable. The vari-
ance of a random variable then quantifies the average squared deviation of a
random variable from its “center”. The Section 2.2 develops these concepts.
The Section 2.3 introduces the notion of the expected value of a general
function of a random variable which leads to the related notions of moments
and moment generating functions (mgf) of random variables. In Section 2.4
we apply a powerful result involving the mgf which says that a finite mgf
determines a probability distribution uniquely. We also give an example which
shows that the finiteness of all moments alone may not determine a probabil-
ity distribution uniquely. The Section 2.5 briefly touches upon the notion of a
probability generating function (pgf) which consequently leads to the idea of
factorial moments of a random variable.

2.2 Expectation and Variance

Let us consider playing a game. The house will roll a fair die. The player
will win $8 from the house whenever a six comes up but the player will pay
$2 to the house anytime a face other than the six comes up. Suppose, for
example, that from ten successive turns we observed the following up
faces on the rolled die: 4, 3, 6, 6, 2, 5, 3, 6, 1, 4. At this point, the player is
ahead by $10(= $24 – $14) so that the player’s average win per game thus
far has been exactly one dollar. But in the long run, what is expected to be
the player’s win per game? Assume that the player stays in the game k
times in succession and by that time the face six appears n times while a
k
non-six face appears k – n times. At this point, the player’s win will then
k
amount to 8n – 2(k – n ) so that the player’s win per game (W ) should
k k k
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