Evaluating E(X) and Var X by moment generating function Xijin Ge SDSU Stat/Math Mysterious Mathematics Ahead! Student Discretion Advised.

E(X) for a geometric r.v. By Definition: Both sides multiply by q: Subtract above equations: Geometric Series!

Var X for a geometric r.v. By Definition: ? ? ? ?

K-th ordinary moment Moment Generating Function provides an “easier” method to calculate E(x), E(x^2) etc If we can work out a mathematical expression for the m.g.f., then we could take derivatives to obtain the ordinary moments 1st ordinary moment 2nd ordinary moment

Proof:

M.G.F. for geometric distribution Geometric series

Using M.G.F. to calculate E(X) for geometric distribution Its first derivative: Evaluating this derivative at t=0:

Using M.G.F. to calculate Variance for geometric distribution Second order derivative: Evaluating it at t=0:

Geometric Distribution Cumulative distribution function: Moment generating function:

M.G.F.’s are wonderful: Completely identifies a distribution. If a distribution has a m.g.f., then it is unique. If a r.v. has a m.g.f.: then it follows a geometric distribution with p=0.4