Moment generating function

In the discrete probability distribution, we studied how to calculate the average. One of the functions for average is the moment generating function.

$g(x) = E(e^{tx})$

In a simple example, we can use the moment generating function.

X	0	1	2
p	1/3	1/3	1/3

In this table, we can add the row for $e^{tx}$ .

$e^{tx}$	e	$e^{t}$	$e^{2t}$
X	0	1	2
p	1/3	1/3	1/3

$E(e^{tx})= \frac{1}{3}+ \frac{1}{3}*e^{t}+ \frac{1}{3}*e^{2t}$

$E(e^{tx}) = \sum_{i} e^{txi} * p(x_{i} )$

Before we look though moment generating function, we should know the definition of the moment.

$Moment = E(X^{n})$

Moment generating function(mgf) is a function that generates "Moment" by taking the derivative of moment generating function(E( $e^{tx}$ )) with respect to t and then inserting 0 in t.

1) Taking the derivative of moment generating function(E( $e^{tx}$ )) with respect to t.

$\frac{d(E(e^{tx})) }{dt}= \frac{d \sum e^{tx_{i}}*p(x_{i}) }{dt} =\frac{ \sum d e^{tx_{i}}*p(x_{i}) }{dt} = \sum p(x_{i}) \frac{ d e^{tx_{i}} }{dt}$ $\sum P(x_{i})\frac{e^{tx_{i}}}{dt} = \sum P(x_{i})*x_{i} *e^{tx_{i}} (\because \frac{e^{t}}{dt} = e^{t}, chain rule \frac{e^{at}}{dt} = a*e^{at} )$

2) Insert 0 in t.

$\sum P(x_{i})*x_{i} *e^{t{x_{i}}} = \sum P(x_{i})*x_{i} = E(X)$

Knowing mgf is important because you can get to know pmf using mgf.

Moment generating function

Posted by Happy Bioinfo

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Moment generating function

Posted by Happy Bioinfo

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