 
Summary: Expectation
Suppose E is an experiment the set of possible outcomes of which is the sample space S. Let E be an
event, which is to say that E S. Let us recall the relative frequency interpretation of P(E), the probability
of E. Let
s1, s2, . . . , sn, . . .
be the outcomes of a never ending sequence of independent repetitions of E. Let (E, n), n = 1, 2 . . . be the
number of occurrences of E in the first n repetitions of E; that is, (E, n) is the number of i {1, . . . , n}
such that si E. Then
P(E) = lim
n
(E, n)
n
.
Now suppose X is a random variable on S with finite range x1, . . . , xN . We define the expectation
E(X) of X to be
lim
n
X(s1) + X(s2) + · · · + X(sn)
n
which is just the limit of the running average values of X on the sequence of outcomes s1, s2, . . . , sn, . . .. I
