Database Reference
In-Depth Information
Mean and Variance
Most people know the expectation, E[X], as the
mean
or the
average
of
X. Using the example from the previous section, the expected (or average)
number of red balls out of 5 can be easily computed. Drawing 0 red balls can
be skipped because 0 times anything is still 0. The remaining 4 possibilities
are then added together:
Multiplying out all the terms and engaging in a lot of tedious canceling of
terms eventually yields 5p as the expected number of red balls when 5 balls
are drawn.
Of course, the actual number of red balls drawn in a particular group of 5
varies and, depending on p is probably not even an integer, which makes
it impossible to draw the “expected” number of red balls. The dispersion of
the observed number of red balls around the expected number of balls is
measured by the
variance
, written Var(X). Most people have encountered
the square root of the variance, which is called the
standard deviation
.
Asithappens, thevariance isactually justanother expectation. Thevariance
is defined as Var(X) = E[(X-E[X])
2
]. This expands to E[X
2
] - (E[X])
2
.
The calculation is the same as for any other expectation and, omitting the
intermediate calculation, Var(X), for the earlier example is 5×p×(1-p).
Other Moments
In addition to the mean and variance being special names for the
expectation of the first two powers of X (or of X-E[X] in the case of the
variance and other higher powers), some of the other powers of X have
special names. In general, these powers of the expectation of X are called
moments
or, if they are powers of X-E[X],
central moments
of X.
These moments come up in later sections of this chapter to help calculate
interesting parameters from observed data. For example, in the example
used in the last few sections, p is a recurring parameter. In the problem
presented, p is the number of red balls in the urn divided by the total
