Properties of the Distribution

Calculation of Mean, Variance, and Moment Generating Function

Calculating E( X ), the expectation or mean of the Geometric( p ) distribution.

> restart;

> with(plots, display):

> f:=x->GeometricPDF(p,x);

> EX:=simplify(sum(x*f(x),x=1..infinity));

Calculating Var( X ), the variance of the Geometric( p ) distribution.

We will employ the formula: Var( X ) = E( ) -

> E_X_SQ:=simplify(sum(x^2*f(x),x=1..infinity));

> VarX:=simplify(E_X_SQ-EX^2);

The Moment Generating Function (MGF) can be easily calculated via Maple.

Recall the moment generating function of a random variable X is defined

as M ( t ) = E( ), provided this expectation exists.

> simplify(sum(exp(t*x)*f(x),x=1..infinity));

So the moment generating function for a Geometric( p ) random variable

is given by

M ( t ) =

We will define M ( t ) as a function of t .

> M:=t->exp(t)*p/(1-exp(t)+exp(t)*p);

The moment generating function provides us alternative ways to calculate the

mean and variance by way of the formula.

(0) = E( ) , where ( t ) denotes the r th derivative of M ( t ) with respect to t .

This formula holds as long as M ( t ) exists in an open interval containing zero.

See, for example, Mathematical Statistics and Data Analysis by John A. Rice for

more on the moment generating function.

> M_p:=diff(M(t),t);

> simplify(M_p);

> simplify(subs(t=0,M_p));

And therefore if X is a Geometric( p ) variable, then E( X ) = 1/ p , which agrees with

what we found earlier. Now turning to the second moment.

> M_pp:=diff(M_p,t);

> simplify(subs(t=0,M_pp));

>

Therefore E( ) = , which again is in agreement with the value

calculated previously. The variance is now quickly calculated as

Var( X ) = E( ) -

=

= .