moment generating functions - METU OCW [PDF]

MOMENT GENERATING FUNCTIONS

1

Moments For each integer k, the k-th moment of X is

  E  X   the k-th moment * k

k

k  E  X     the k-th central moment k

2

MOMENT GENERATING FUNCTION (mgf) • Let X be a rv with cdf FX(x). The moment generating function (mgf) of X, denoted by MX(t), is

  etx p ( x) , if X is discrete  all x tX M X  t   E e    tx e f  x  dx, if X is continuous   all x

provided that expectation exist for t in some neighborhood of 0. That is, there is h>0 such that, for all t in h