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Moment-generating Function
(Still technically lecture #6 but very different topic)
X := real r.v.
M_X(t) = \exp e^{tX} where t \in R.
Defined if \int_{-\infty}^\infty e^{tx} f_x(x) dx < \infty
for t \in (-h, h) for some h > 0. [I can't remember why the region
of convergence is symmetric about 0, but I remember some thm. about
that]
e^{tx} gives a nice Taylor series.
For M_X(t) around 0,
M_X(t) = M_X(0) + M_X'(0) t + M_X''(0)t^2/2 + M_X'''(0) t^3/3! + ...
M_X^{(k)}(t) = {d^k\over dt^k} \exp{e^{tX}} = {d^k\over dt^k}
\int_{-\infty}^\infty e^{tx} f_x(x) dx
= \int_{-\infty}^\infty x^k e^{tx} f_x(x) dx
= \exp[X^k e^{tX}]
= [with t = 0] \exp[X^k].
Why is it useful?
Example: X ~ N(\mu, \sigma^2) *can* have its moments computed by
integration-by-parts (probably table method), but the mgf can be used
instead, which makes the determination easier.
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