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properties of expectation

Note: Linearity is special!It is not true in general that

E[X•Y] = E[X] • E[Y]E[X2] = E[X]2

E[X/Y] = E[X] / E[Y]E[asinh(X)] = asinh(E[X])

•••

variance

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Van x E X ENDY EAT ECxX standarddeviation He

VarfaXtb a2Var X a b are constants

more variance examples

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-4 -2 0 2 4

0.00

0.10

-4 -2 0 2 4

0.00

0.10

-4 -2 0 2 4

0.00

0.10

-4 -2 0 2 4

0.00

0.10

0.20

σ2 = 5.83

σ2 = 10

σ2 = 15

σ2 = 19.7

p Pr Xix X

Xx

Xy

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Van EGA EDFVan XH f Van X Van Y in general

ExampleI

f Ea ears I

Y X F Y 0

Vanly L

Van Vav 2 f Van XH O

ExaVan Xix Van 2X 2 Van 4 Vana

f Vallytvalx

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Random variables independence

Arau X and an event E are independent fFx Pr X x NE Pr X x PrlE

2r.vn's X Y are independent ofKitty Pr X xnY y PrfX x PrCY y

Flip in independently 2h timesZn heads in 2h tresses

Xi heads in Austin tosses

ya He heads in 2nd n tosses

X Y indep X Z notindep

pr x i find 7 I prfx.u.az fPrfX x Prl2

4PrfYj lngn1 t.FM O ZenPrix.im j 7 j Lat 2h

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Thmi If X Y are independent then E XuY E X E Y

F XY ab Pr n'Kb

a b Prcx a Prats bgk.LK

aPrCX a bPrCY b

IEH EG

Z I 24 2 7

9

Theorem: If X & Y are independent, then Var[X+Y] = Var[X]+Var[Y]

Proof:

variance of independent r.v.s is additive

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a zoo of (discrete) random variables

discrete uniform random variables

A discrete random variable X equally likely to take any (integer) value between integers a and b, inclusive, is uniform.

Notation:

Probability mass function:

Mean:

Variance:

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Toss die 1,2 as a at at2 a b9 1 b 6

Uniffa b it'ra b3

Pr X i o itta is

are2

b a b at2

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discrete uniform random variables

A discrete random variable X equally likely to take any (integer) value between integers a and b, inclusive, is uniform.

Notation: X ~ Unif(a,b)

Probability:

Mean, Variance:

Example: value shown on one roll of a fair die is Unif(1,6):

P(X=i) = 1/6 E[X] = 7/2 Var[X] = 35/12

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0 1 2 3 4 5 6 7

0.10

0.16

0.22

i

P(X=i)

In a series X1, X2, ... of Bernoulli trials with success probability p, let Y be the index of the first success, i.e., X1 = X2 = ... = XY-1 = 0 & XY = 1Then Y is a geometric random variable with parameter p.

Examples:

Number of coin flips until first head

Number of blind guesses on SAT until I get one right

Number of darts thrown until you hit a bullseye

Number of random probes into hash table until empty slot

Number of wild guesses at a password until you hit it

Probability mass function:

Mean: Variance:

geometric distribution

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X n GeoCp

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i Ip p i ppr Xo owL

pp2

In a series X1, X2, ... of Bernoulli trials with success probability p, let Y be the index of the first success, i.e., X1 = X2 = ... = XY-1 = 0 & XY = 1Then Y is a geometric random variable with parameter p.

Examples:

Number of coin flips until first head

Number of blind guesses on SAT until I get one right

Number of darts thrown until you hit a bullseye

Number of random probes into hash table until empty slot

Number of wild guesses at a password until you hit it

P(Y=k) = (1-p)k-1p;

Mean 1/p; Variance (1-p)/p2

geometric distribution

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Bernoulli random variables

An experiment results in “Success” or “Failure”X is an indicator random variable (1 = success, 0 = failure) P(X=1) = p and P(X=0) = 1-pX is called a Bernoulli random variable: X ~ Ber(p)

Mean:

Variance:

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indicator riv

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P O O l p

P Ph E ExPrCXp tp

xeRangelx

Vancx E XY Efx 2Pp

Bernoulli random variables

An experiment results in “Success” or “Failure”X is an indicator random variable (1 = success, 0 = failure) P(X=1) = p and P(X=0) = 1-pX is called a Bernoulli random variable: X ~ Ber(p)E[X] = E[X2] = pVar(X) = E[X2] – (E[X])2 = p – p2 = p(1-p)

Examples:coin fliprandom binary digitwhether a disk drive crashed

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Jacob (aka James, Jacques) Bernoulli, 1654 – 1705