From 32f4af5f369fa9f0b2988ecad7797f4bec3661c3 Mon Sep 17 00:00:00 2001
From: Holden Rohrer <hr@hrhr.dev>
Date: Tue, 21 Sep 2021 17:12:46 -0400
Subject: notes and homework

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 zhilova/07_mgf | 22 ++++++++++++++++++++++
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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.
-- 
cgit