Concrete mathematics : a foundation for computer science

8.1 DEFINITIONS 371
making independent trials in such a way that each value of X occurs with
a frequency approximately proportional to its probability. (For example, we
might roll a pair of dice many times, observing the values of S and/or P.) We’d
like to define the average value of a random variable so that such experiments
will usually produce a sequence of numbers whose mean, median, or mode is
approximately the s,ame as the mean, median, or mode of X, according to our
definitions.
Here’s how it can be done: The mean of a random real-valued variable X
on a probability space n is defined to be
t x.Pr(X=:x) (8.6)
XEX(cl)
if this potentially infinite sum exists. (Here X(n) stands for the set of all
values that X can assume.) The median of X is defined to be the set of all x
such that
Pr(X6x) 3 g and Pr(X3x) 2 i. (8.7)
And the mode of X is defined to be the set of all x such that
Pr(X=x) 3 Pr(X=x’) for all x’ E X(n). (8.8)
In our dice-throwing example, the mean of S turns out to be 2. & + 3.
$ +... + 12. & = 7 in distribution Prcc, and it also turns out to be 7 in
distribution Prr 1. The median and mode both turn out to be (7) as well,
in both distributions. So S has the same average under all three definitions.
On the other hand the P in distribution Pro0 turns out to have a mean value
of 4s 4 = 12.25; its median is {lo}, and its mode is {6,12}. The mean of P is
unchanged if we load the dice with distribution Prll , but the median drops
to {8} and the mode becomes {6} alone.
Probability theorists have a special name and notation for the mean of a
random variable: Th.ey call it the expected value, and write
EX = t X(w) Pr(w).
wEn
(8.9)
In our dice-throwing example, this sum has 36 terms (one for each element
of !J), while (8.6) is a sum of only eleven terms. But both sums have the
same value, because they’re both equal to
1 xPr(w)[x=X(w)]
UJEfl
XEX(Cl)