Statistical Methods in Medical Research 4ed

3.5 Expectation
There is some difficulty in deciding what is meant by the phrase `mean of a
random variable'. The mean has been defined earlier only for a finite number, n,
of observations. With a probability distribution such as that in Table 3.1 the
number of observations must be thought of as infinite. How, then, is the mean to
be calculated?
Suppose n is very largeÐso large that the relative frequencies of the different
values of a discrete random variable like that in Table 3.1 can be taken to be very
nearly equal to the probabilities. If they were exactly equal to the probabilities,
the frequency distribution of the number of boys would be as follows:
The mean value of x would be
x
No. of boys Frequency
0 0 0576n
1 0 2400n
2 0 3747n
3 0 2600n
4 0 0677n
…0 0 0576n†‡…1 0 2400n†‡…2 0 3747n†‡…3 0 2600n†‡…4 0 0677n†
:
n
The factor n may be cancelled from the numerator and denominator of the
expression, to give the numerical result 2 04. The arbitrary sample size, n, does
not appear.
If the probabilities of 0, 1, ..., 4 boys are denoted by P0, P1, ..., P4, the
formula for the mean is clearly
…0 P0†‡…1 P1†‡…2 P2†‡…3 P3†‡…4 P4†:
In general, if x is a discrete random variable taking values x0, x1, x2, ..., with
probabilities P0, P1, P2, ..., the mean value of x is calculated as
P
xiPi: …3:6†
i
The mean value of a random variable, calculated in this way, is often called the
expected value, the mathematical expectation or simply the expectation of x, and
the operation involved (multiplying each value of x by its probability, and then
adding) is denoted by E(x). The expectation of a random variable is often
allotted a Greek symbol like m (lower-case Greek letter `mu'), to distinguish it
from a mean value calculated from a finite number of observations (denoted
usually by symbols such as x or y).
n
3.5 Expectation 63