**Mumbai University > Electronics and Telecommunication Engineering > Sem 5 > Random Signal Analysis

Marks: 10M

Year: May 2016

The distribution of a random variable is often characterized in terms of its moment generating function (mgf), a real function whose derivatives at zero are equal to the moments of the random variable.

Let X is a random variable. If the expected value E(e^θX ) exists and is finite for all real numbers θ belonging to a closed interval [-h,h]⊂R with h>0, then we say that X possesses a moment generating function and the function

$M_X$ (θ) = E($e^θ)^X$ )

is called the moment generating function of X.

∴ $M_X$ (θ)= E($e^θ)^X$ )=∑$(e^θ)^X$. P(X=x) if X is discrete.

$M_X$ (θ)=E$(e^θ)$X )=$ \int \limits_{ \infty}^{- \infty} F_{X} (x) dx$ if X is continuous

for all real θ for which the sum or integral converges absolutely.

Moments about the origin may be found by power series expansion: thus we may write

$M_X$ (θ)=$E {(e^{θX})}$

=E($\sum \limits_{r=0}^{\infty}(θX)^r/r!)$

=($\sum \limits_{r=0}^{\infty}θ^r E(X^r)/r!)$

$$i.e.$$

$M_X (θ)$=($\sum \limits_{r=0}^{\infty}(θ)^r.μ_{r}'/r!)$ where $μ_r'$=E($X^r$)