Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 811 — #819
19.4. Great Expectations 811
0:25
0:2
0:15
f 20;:75 .k/
0:1
0:05
0
0
5
10 15 20
k
Figure 19.5 The pdf for the general binomial distribution f n;p .k/ for n D 20
and p D :75.
19.4 Great Expectations
The expectation or expected value of a random variable is a single number that reveals
a lot about the behavior of the variable. The expectation of a random variable
is also known as its mean or average. For example, the first thing you typically
want to know when you see your grade on an exam is the average score of the
class. This average score turns out to be precisely the expectation of the random
variable equal to the score of a random student.
More precisely, the expectation of a random variable is its “average” value when
each value is weighted according to its probability. Formally, the expected value of
a random variable is defined as follows:
Definition 19.4.1. If R is a random variable defined on a sample space S, then the
expectation of R is
ExŒR WWD X !2S
R.!/ PrŒ!: (19.2)
Let’s work through some examples.