Properties of the Distribution

Calculation of Mean, Variance, and Moment Generating Function

To calculate the mean of the Normal distribution, we can simply integrate x *(NormalPDF), per the definition of mathematical expectation.

> restart;

> with(plots, display):

> interface(showassumed=0):

> assume(sigma > 0);

> f(x):=NormalPDF(mu,sigma^2,x);

> EX:=int(x*f(x),x=-infinity..infinity);

Calculating Var( X ), the variance of the Normal( , ) distribution. We will employ the formula: Var( X ) = E( ) -

> E_X_SQ:=int((x^2)*f(x),x=-infinity..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.

> int(exp(t*x)*f(x),x=-infinity..infinity);

So the moment generating function for a Normal( , ) random variable is given by

M ( t ) =

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

> restart; with(plots):

> M(t):=exp(mu*t+(sigma^2*t^2)/2);

The moment generating function provides us with 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. Looking at the first derviative, we find:

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

> simplify(M_p);

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

This agrees with our previous result, . Turning to the second derivative, we find:

> 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( ) -

=

=