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Discrete random variables

Moments. The mean (or expectation) of X is sometimes called the first moment of X, since it
is defined as the sum or integral of the probability density function multiplied by the first power of
x. By a simple extension the r-th moment of a distribution is defined by

{
∑ r
x f(x j ) for a discrete distribution ,
r
µ r ≡ E(X ) = ∫ j j (5.9)
r
x f(x)dx for a continuous distribution .
r
For notational convenience, we have introduced the symbol µ r to denote E(X ), the r-th
moment of the distribution. Clearly, the mean of the distribution is then denoted by µ 1 , often
abbreviated simply to µ, as in the previous subsection, as this rarely causes confusion. A useful
result that relates the second moment, the mean and the variance of a distribution is proved
using the properties of the expectation operator:

2
2
2
2
2
Var(X) = E[(X − µ) ] = E(X − 2µX + µ ) = E(X − 2µE(X) + µ ) =
2
2
2
2
2
= E(X ) − 2µ + µ = E(X ) − µ . (5.10)
Example 5.4. A biased die has probabilities p/2, p, p, p, p, 2p of showing 1, 2, 3,
4, 5, 6 respectively. Find the mean, the second moment and the variance of this
probability distribution. ,
Solution. By demanding that the sum of the probabilities equals unity we require p = 2/13.
Now, using the definition of the mean (5.6) for a discrete distribution,

∑ 1
E(X) = x j f(x j ) = 1 · p + 2 · p + 3 · p + 4 · p + 5 · p + 6 · 2p =
2
j
53 53 2 53
= p = × = .
2 2 13 13

Similarly, using the definition of the second moment (5.9),

∑ 1 253 253
2
2
2
2
2
2
2
2
E(X ) = x f(x j ) = 1 × p + 2 p + 3 p + 4 p + 5 p + 6 × 2p = p = .
j 2 2 13
j
Finally, using the definition of the variance (5.7), with µ = 53/13, we obtain
∑ 1
2
2
2
2
Var(X) = (x j − µ) f(x j ) = (1 − µ) 2 p + (2 − µ) p + (3 − µ) p + (4 − µ) p+
2
j
3120 480
2
2
+(5 − µ) p + (6 − µ) 2p = p = .
169 169
2
2
It is easy to verify that Var(X) = E(X ) − (E[X]) .
Central moments. The variance Var(X) is sometimes called the second central moment of the
distribution, since it is defined as the sum or integral of the probability density function multiplied
by the second power of x−µ. The origin of the term ’central’ is that by subtracting µ from x before
squaring we are considering the moment about the mean of the distribution, rather than about
x = 0. Thus the k-th central moment of a distribution is defined as
{
∑ k
(x j − µ) f(x j ) for a discrete distribution,
k
ν k = E((X − µ) ) = ∫ j (5.11)
k
(x − µ) f(x)dx for a continuous distribution.
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