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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: et et 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.