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GATech - MATH 3670 - Class Notes - Week 5

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Mathematical expectation of a n.v. XIf you want to learn more check out What are the defining characteristics of coal mining?

Denoted by EX, given (when it exists)

By EX = xPx(x)

= CPx(x) = CRx(x) = C1 = CDon't forget about the age old question of What are the Berber influences from the North?

E(dx) = d E(x)If you want to learn more check out Describe the yellow river cultures.

E(dx) = dxPx(x) = dEx d ERWe also discuss several other topics like What is the Signal Detection Theory?

E(x + B) = x + B Px(x)

= xPx(x) + BPx(x) = xPx(x) + BPx(x)We also discuss several other topics like What is the difference between rote counting and rational counting?

E(x) + BDon't forget about the age old question of What are numbers calculated by comparing a number to a base number?

“Expectation is Linear”

E(dx +B) = dE(x) + B Vd, BER

In general it is not true that E(xy) = EXEY

Hint: to find a counterexample to the above equity, take x = y

Take X such that x = +1 w.p. -1 w.p.

Then E(x) = 1. + -1. = 0

Take x = y then xy = x2 = 1

So E(xy) = E(x2) =1 EXEY = 0.0 = 0

Note: Let x be a n.v. and let f:R → R be a function

Then f(x) is also a n.v.

f(x) = f0x

So f0x is a function from to R ie f0x is a n.v.

If it exists

Ef(x) = f(x) P(x = x)

f(x) Px(x)

Existence for us is going to mean that |f(x)|Px(x) < +

Definition: let x be a n.v. if it exists, the variance of x denoted by varX is given by: varX = E[(x - Ex)2] = E(x2) - (EX)2

Example: Let X, Y, and Z be three distinct n.v. respectively given by

X = 0, y = +1 w.p. -1 w.p. and Z = +10 w.p. -10 w.p.

Then EX = 0, EY = 1. + (-1). = 0, EZ = 10.+ (-1).) = 0

Var X = E[(E - EX)2] = E[x - 0]2 = E(x2) = 0 var Y = E[(y - EY)2] = E(12) = E(1) = 1

Var Z = E[(z - EZ)2] = E(102) = E(100) = 100

The higher the valence, the more spread out a random variable is vis a vis its expectation

Properties of variance

- If x = c, then var X = 0, indeed

Var X = (x - EX)2 px(x)

= (C - C)2 Px(x) = OPx(x) = 0

- Var (∝X) = (dx - E(dx)2Px(x)

= ∝2 var X

- Var (x + B) = (x + B - E(x + B))2 Px(x)

(x + B - EX - B)2 Px(x)

= var X

- In general, var (x + y ) var X + var Y

Indeed taking x = y we have

var(X + Y) = var(2x) = 4 var X var X + var Y = 2 var X

- Var X = 0 < = > (x - EX)2 Px(x) = 0

So var x = 0 < = > x - EX = 0

Definition: the standard deviation of a n.v. X is the positive square root of the variance, of O3x denotes the variance of X, so 6x denotes the standard deviation of X, in particular O2x = |∝}Ox and 6x + B = 6x