Page 168 - DMTH404_STATISTICS

Statistics

Notes

Notes that the above integral is finite for t  [–h, h] for any 0 < h < , so that X possesses
a moment generating function.


M (t) =
X   t

12.2 Deriving Moments with the mgf

The moment generating function takes its name by the fact that it can be used to derive the
moments of X, as stated in the following proposition.
Proposition If a random variable X possesses a moment generating function M (t), then, for any
X
n , the n-th moment of X (denote it by  (n)) exists and is finite.
X
Furthermore:

n
d Mx(t)
 (n) = E[X ] =
n
X n
dt

t 0
n
d Mx(t)
where is the nth derivative of M (t) with respect to t, evaluated at the point t = 0.
dt n X

t 0
Providing the above proposition is quite complicated, because a lot of analytical details must be
taken care of (see e.g. Pfeiffer, P.E. (1978) concepts of probability theory, Courier Dover
Publications). The intuition, however, is straightforward: since the expected value is a linear
operator and differentiation is a linear operation, under appropriate conditions one can
differentiate through the expected value, as follows:
n
d M (t)  d n E[exp(tX)] E  d n exp(tX)  E[X exp(tX)]

n
X

dt n dt n   dt n  
which, evaluated at the point t = 0, yields
n
d M (t)  E[X exp(0.X)] E[X ]   (n)
X
n
n

dt n X
t 0

Example: Continuing the example above, the moment generating function of an
exponential random variable is

MX(t) =
  t
The expected value of X can be computed by taking the first derivative of the moment generating
function.
dM (t)  l
X
dt (  t) 2

and evaluating it at r = 0.

160 LOVELY PROFESSIONAL UNIVERSITY