Expectation value

Creator

Creator

Created

Created

2023 Mar 9 1:27

Editor

Editor

Edited

Edited

2025 Jan 29 23:12

Refs

Refs

Average, Mean

Expectation operator is linear

E[X] = \Sigma_x{xP(X= x)} = \int{xf(x)dx}

E_X[g(x)] = \int_\infty^{-\infty}g(x) f_X(x)dx

k-th moment of X

defined as the expected value of x raised to the power of k

E[X^k] =\Sigma_xx^kP(X = x)

If only X, Y is independence,

E[XY] = E[X]E[Y]

E[X]=0

does not means

E[X^2] = 0

Expectation value Notion

Law of total expectation

The law of Large number

notion image

Averages

Arithmetic Mean

Geometric Average

Arithmetic Mean and

Harmonic Mean are Pythagorean means

Pythagorean means

In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.

Pythagorean means

https://en.wikipedia.org/wiki/Pythagorean_means

Pythagorean means

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