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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}
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