notes and homework

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+    Expectation/Expected Value/Mean Value/Average of an r.v.:
+    (Does not exist for all r.v.)
+We must assume that \int_{-\infty}^\infty |x|f_x(x) dx < \infty, so
+
+E(X) := \int_{-\infty}^\infty xf_x(x) dx
+= {\bb E} X = E X.
+
+If discrete,
+E(X) = \sum_{x\in D} xp_x(x)
+
+    Higher (order) moments of X
+moment of kth order := {\bb E}(X^k)
+Again, they do not always exist, but they do exist if {\bb E}(|X^k|)
+exists.
+
+    Variance/dispersion of X
+Var(X) = {\bb E}(X - {\bb E} X)^2
+aka quadratic deviation
+\def\exp{{\bb E}}
+
+Thm: [ proof in textbook ] (1)
+g : R \mapsto R.
+
+Let \int |g(x)| f_x(x) < \infty
+Therefore, \exp g(X) = \int_{-\infty}^\infty g(x)f_x(x) dx
+
+Ex:
+    \exp X^2 = \int x^2 f_x(x) dx
+    \exp(X-a) = \int (x-a) f_x(x) dx
+    \exp\sin X = \int sin x f_x(x) dx
+
+    Stdev
+Stdev := \sqrt{Var(x)}
+
+    Properties of E(x)
+1) Linearity
+    Where E(X), E(Y) exist, and a, b \in R
+    E(aX + bY) = aE(X) + bE(Y)
+    By thm (1), \int axf_x(x) dx = a \int xf_x(x) dx.
+2) E(a) = a
+3) If g(x) \geq 0, E(g(X)) \geq 0, regardless of X.
+
+Example application:
+Var(X)
+= E [X - E[X]]^2
+= E [ X^2 - 2X * E[X] + [E[X]]^2 ]
+= E[X^2] - 2E[X]^2 + [E[X]]^2
+           ^ linearity applied with E[X] as constant
+= E[X^2] - E[X]^2
+
+On the reals (by property 3),
+Var(X) \geq 0
+\to E(X^2) - E(X)^2 \geq 0
+\to E(X^2) \geq E(X)^2 [equality is strict unless X = a]
+
+More example:
+Var(aX) = E[aX]^2 - (E[aX])^2
+        = E[a^2X^2] - (aE[X])^2
+        = a^2E[X^2] - a^2E[X]^2
+        = a^2(Var(X))
+
+Definitions:
+1) centering: X - \exp X. \exp[X - \exp X] = 0.
+2) rescaling: With c>0, cX. Var(cX) = c^2 Var X.
+3) centering and standardization: centering and rescaling s.t.
+Var(Y) = 1.
+    Y = (X - \exp X)/\sqrt{Var X}