Page 182 - DMTH404_STATISTICS

Statistics

Notes  When t = 0, it clearly follows that M(0) = 1. Now by differentiating M(t)r times, we obtain

d r  d t tX 
tX
r tX
(r)
M (t) = E[e ] E  e   E[X e ]

dt r  dt r 
In particular when t = 0, M (0) generates the r-th moment of X as follows.
(r)
r
M (r)(0) = E[X ], r = 1, 2, 3,...
13.4 Keywords
Moment-generating function: In probability theory and statistics, the moment-generating function
of any random variable is an alternative definition of its probability distribution.

Standard normal random variable: The moment generating function M (t ) of a standard normal
Xi i
random variable is defined for any t  . As a consequence, the joint moment generating
i
K
function of X is defined for any t   .
13.5 Self Assessment

1. In addition to ................., moment-generating functions can be defined for vector- or matrix-
valued random variables, and can even be extended to more general cases.
(a) finite cross-moments (b) uniquely determines

(c) univariate distributions, (d) moment-generating
2. The ................. function does not always exist even for real-valued arguments, unlike the
characteristic function. There are relations between the behavior of the moment-generating
function of a distribution and properties of the distribution, such as the existence of moments.
(a) finite cross-moments (b) uniquely determines
(c) univariate distributions, (d) moment-generating

3. If a K × 1 random vector X possesses a joint moment generating function M (t), then, for
X
any n  , X possesses ................. of order n.
(a) finite cross-moments (b) uniquely determines

(c) univariate distributions, (d) moment-generating
4. The most significant propertyof moment generating function is that “the moment
generating function ................. the distribution.”

(a) finite cross-moments (b) uniquely determines
(c) univariate distributions, (d) moment-generating

13.6 Review Questions

1. Let X be a K × 1 standard multivariate normal random vector. Its support R is
X
R =  K
X
and its joint probability density function f (x) is
X

T 
f (x) (2 )  K /2 exp  1 x x 



X
 4 
174 LOVELY PROFESSIONAL UNIVERSITY