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Math 301 Probability and Statistics I – Dr. McLoughlin’s Class Handout II 1/2, page 1 of 3
MATH 301 PROBABILITY AND STATISTICS I
DR. MCLOUGHLIN’S CLASS
MOMENT GENERATING FUNCTIONS (MGFS) FOR RANDOM VARIABLES
HANDOUT 2 ½
Definition 2.5.1: Let X be a discrete random variable where X ~ fX(x). The moment generating
function (MGF) of X [where it exists] is defined where   > 0   t  (-, )
MX(t) = E[ e tX ] =
 etx  fX (x)
x
MX: 
Theorem 2.5.1: Let X be a discrete random variable where X ~ fX(x), ,   ,
and
  > 0   t  (-, ) (where of course , t  ) MX(t) exists.
(1) M(X + )(t) = e t  M X (t)
(2) M(X)(t) = MX (t)
(3) M X +   (t) =





t
t
e   MX ( )

Theorem 2.5.2: Let X be a discrete random variable
Then  a unique function, fX(x), which is its probability mass function where such exists;
 a unique function, FX(x), which is its cumulative distribution function where such exists; and,
Then  a unique function, MX(t), which is its moment generating function where such exists.
We claim there are some Special MGFs associated with discrete random variables:

X ~ Bin (x, p, n)
MX(t) = (1  p)  pet
X ~ Pois(x,)
t
MX(t) = e(e 1)
X ~ Geo(x,p)
MX(t) =
pe t
1  e t (1  p)

n
 t <  ln(1  p)
Math 301 Probability and Statistics I - McLoughlin’s Class Handout 2½ , page 2 of 3
Definition 2.5.2: Let X be a continuous random variable where X ~ fX(x). The moment
generating function (MGF) of X [where it exists] is defined where   > 0   t  (-, )
MX(t) = E[ e tX ] =

e tx  f X (x)dx
x
Theorem 2.5.3: Let X be a continuous random variable where X ~ fX(x), ,   ,
and
  > 0   t  (-, ) (where of course , t  ) MX(t) exists.
(1) M(X + )(t) = e t  M X (t)
(2) M(X)(t) = MX (t)
(3) M X +   (t) =





t
t
e   MX ( )

Theorem 2.5.4: Let X be a continuous random variable
Then  a unique function, fX(x), which is its probability density function where such exists;
 a unique function, FX(x), which is its cumulative distribution function where such exists; and,
Then  a unique function, MX(t), which is its moment generating function where such exists.
We claim there are some Special MGFs associated with continuous random variables:
et  et
t(  )
X ~ Uni(x, , )
MX(t) =
X ~ Nor(x, , )
1
(t  2 t 2 )
2
MX(t) = e
X ~ (x, , )
MX(t) = (1  t)
t<
1

X ~ Exp(x,)
MX(t) = (1  t)1
t<
1

X~Chi(x, )
MX(t) = (1  2t)( / 2)
t<2
Math 301 Probability and Statistics I - McLoughlin’s Class Handout 2½ , page 3 of 3
Theorem 2.5.5: Let X be a random variable with the well defined function, MX(t), which is its
moment generating function such that   > 0   t  (-, ) (where of course , t  ) MX(t)
exists. It is the case that E Xn   M X(n) (t) where M X(n ) (t) is the nth derivative of MX(t) (then.
t 0
obviously, it is evaluated at t = 0).
Last revised : 27 Nov. 2010, © 1998 – 2010, M. P. M. M. M.