MSc. Econ: MATHEMATICAL STATISTICS, 1996

The Moment Generating Function of the Normal Distribution

Recall that the probability density function of a normally distributed random variable x with a mean of E(x)=and a variance of V (x)=2 is

2 1 1 (x)2/2 (1) N(x; , )=p e 2 . (22)

Our object is to nd the moment generating function which corresponds to this distribution. To begin, let us consider the case where = 0 and 2 =1. Then we have a standard normal, denoted by N(z;0,1), and the corresponding moment generating function is dened by Z zt zt 1 1 z2 Mz(t)=E(e )= e √ e 2 dz (2) 2 1 t2 = e 2 .

To demonstate this result, the exponential terms may be gathered and rear- ranged to give exp zt exp 1 z2 = exp 1 z2 + zt (3) 2 2 1 2 1 2 = exp 2 (z t) exp 2 t .

Then Z 1t2 1 1(zt)2 Mz(t)=e2 √ e 2 dz (4) 2 1 t2 = e 2 , where the nal equality follows from the fact that the expression under the integral is the N(z; = t, 2 = 1) probability density function which integrates to unity. Now consider the moment generating function of the Normal N(x; , 2) distribution when and 2 have arbitrary values. This is given by Z xt xt 1 1 (x)2/2 (5) Mx(t)=E(e )= e p e 2 dx (22)

Dene x (6) z = , which implies x = z + .

1 MSc. Econ: MATHEMATICAL STATISTICS: BRIEF NOTES, 1996

Then, using the change-of-variable technique, we get Z 1 1 2 dx t zt p 2 z Mx(t)=e e e dz 2 dz Z (2 ) (7) t zt 1 1 z2 = e e √ e 2 dz 2 t 1 2t2 = e e 2 ,

Here, to establish the rst equality, we have used dx/dz = . The nal equality follows in view of the nal equality of equation (2) above where the dummy variable t can be replace by t. The conclusion is that

(8) The moment generating function corresponding to the normal 2 probability density function N(x; , ) is the function Mx(t)= exp{t + 2t2/2}.

The notable characteristic of this function is that it is in the form of an exponential. This immediately implies that the sum of two independently dis- tributed Normal random variables is itself a normally distributed random vari- able. Thus:

2 2 (9) Let x N(x,x) and y N(y,y) be two independently dis- tributed normal variables. Then their sum is also a normally dis- 2 2 tributed random variable: x + y = z N(x + y,x +y).

To prove this we need only invoke the result that, in the case of indepen- dence, the moment generating function of the sum is the product of the moment generating functions of its elements. This enables us to write 1 2 2 1 2 2 Mz(t)=Mx(t)My(t) = exp xt + 2 xt exp yt + 2 yt (10) 1 2 2 2 = exp (x + y)t + 2 (x + y)t which is recognised as the moment generating function of a normal distribution.

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