Page 101 - Probability and Statistical Inference

78 2. Expectations of Functions of Random Variables

p > 1, let us write , a positive and finite real number, and
define a random variable X which takes the values i ∈ χ = {1, 2, 3, ...} such
that P(X = i) = , i = 1, 2, ... . This is obviously a discrete probabil-
ity mass function. Now, let us fix p = 2. Since is not finite, it
is clear that η or E(X) is not finite for this random variable X. This example
1
shows that it is fairly simple to construct a discrete random variable X for
which even the mean µ is not finite. !
The Example 2.3.1 shows that the moments of a random
variable may not be finite.

Now, for any random variable X, the following conclusions should be fairly
obvious:

The part (iii) in (2.3.1) follows immediately from the parts (i) and (ii).

th
The finiteness of the r moment η of X does not necessarily
r
imply the finiteness of the s moment η of X when s > r.
th
s
Look at the Example 2.3.2.
Example 2.3.2 (Example 2.3.1 Continued) For the random variable X de-
fined in the Example 2.3.1, the reader should easily verify the claims made in
the adjoining table:

Table 2.3.1. Existence of Few Lower Moments
But Non-Existence of Higher Moments
p Finite η Infinite η
r r
2 none r = 1, 2, ...
3 r = 1 r = 2, 3, ...
4 r = 1,2 r = 3, 4, ...
k r = 1, ..., k – 2 r = k – 1, k, ...

The Table 2.3.1 shows that it is a simple matter to construct discrete
random variables X for which µ may be finite but its variance σ may not
2
be finite, or µ and σ both could be finite but the third moment η may not
2
3
be finite, and so on. !