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Explain what is moment generating function and why is it called so?

Mumbai University > Electronics and Telecommunication > Sem5 > Random Signal Analysis

Marks: 4M

Year: Dec 2014

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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.

$$M_X (θ)=E(e^{θX} )$$

is called the moment generating function of X.

$$∴ M_X (θ)=E(e^{θX} )=\sum_xe^{θx}.P(X=x)$$ if X is discrete

$$M_X (θ)=E(e^{θX} )=∫_∞^{-∞}e^{θx} f_X (x)dx$$ 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

$$M_X (θ)=E(e^{θX} )$$

$$=E(\sum_{r=0}^∞\frac{(θX)^{r}}{r!} )$$

$$=(\sum_{r=0}^∞\frac{(θ)^{r}}{r!} E{X^r} )$$

i.e.

$$M_X (θ)=(\sum_{r=0}^∞\frac{(θ)^r}{r!}.μ_r' )$$

where $μ_r'=E(X^r)$