properties of expectation Note: Linearity is special! It is not - PowerPoint PPT Presentation

properties of expectation Note: Linearity is special! It is not true in general that E[ X•Y ] = E[ X ] • E[ Y ] E[ X 2 ] = E[ X ] 2 E[ X/Y ] = E[ X ] / E[ Y ] E[asinh( X )] = asinh(E[ X ]) • • • ! 1

variance E Cx X ENDY EAT Van x E He X standard deviation a2Var X are constants a b VarfaXtb ! 2

more variance examples σ 2 = 5.83 0.10 X Pr Xix p 0.00 -4 -2 0 2 4 σ 2 = 10 0.10 Xx 0.00 -4 -2 0 2 4 σ 2 = 15 0.10 0.00 -4 -2 0 2 4 0.20 σ 2 = 19.7 Xy 0.10 0.00 -4 -2 0 2 4 ! 3

Van EDF EGA in general Van X Van Y Van XH f Example I f Ea ears I Y X 0 F Y L Vanly Van XH O f 2 Van Vav Exa Van Van 2 Van 4 Vana Xix 2X f Vallytvalx ! 4

Random variables independence are independent f an event E Arau X and x PrlE Pr X Pr X Fx NE x of 2r.vn's X Y are independent PrfX x PrCY y Pr X xnY y Kitty 2h times Flip independently in 2h tresses heads in Zn tosses Austin heads in Xi tosses in 2nd n He heads ya Y indep X X Z not indep 7 x Prl2 I fPrfX find 4PrfY prfx.u.az pr x i Zen t.FM j O lngn1 7 j Lat 2h Prix.im j ! 5

then E XuY X Thmi Y If E X E Y are independent n'Kb Pr ab F XY Prats bgk.LK a b Prcx a b bPrCY aPrCX a EG EH I Z 2 I 7 2 4 9 ! 6

variance of independent r.v.s is additive Theorem: If X & Y are independent, then 
 Var[X+Y] = Var[X]+Var[Y] Proof: 
 ! 10

a zoo of (discrete) random variables ! 11


 discrete uniform random variables A discrete random variable X equally likely to take any (integer) value between integers a and b , inclusive, is uniform. Toss die 1,2 as a b at at2 a b 9 1 6 it'ra b3 Notation: Uniffa b itta is Pr X i Probability mass function: o are Mean: 2 Variance: b a b at2 12 ! 12

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 
 0.22 roll of a fair die is Unif(1,6): P(X=i) 0.16 P( X=i ) = 1/6 
 E[ X ] = 7/2 
 0.10 Var[ X ] = 35/12 0 1 2 3 4 5 6 7 ! 13 i

geometric distribution with P upht cointosses In a series X 1 , X 2 , ... of Bernoulli trials with success probability p, let Y be the index of the first success, i.e., X 1 = X 2 = ... = X Y-1 = 0 & X Y = 1 Then Y is a geometric random variable with parameter p. n GeoCp X Examples: Number of coin flips until first head TT Number of blind guesses on SAT until I get one right Number of darts thrown until you hit a bullseye Chp 4p Number of random probes into hash table until empty slot Number of wild guesses at a password until you hit it i pr X Probability mass function: p Ip i p o ow L Mean: Variance: p p2 ! 14

geometric distribution In a series X 1 , X 2 , ... of Bernoulli trials with success probability p, let Y be the index of the first success, i.e., X 1 = X 2 = ... = X Y-1 = 0 & X Y = 1 Then 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-1 p; Mean 1/p; Variance (1-p)/p 2 ! 15


 Bernoulli random variables indicator riv 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-p X is called a Bernoulli random variable: X ~ Ber(p) I II 4 I expectation s p Mean: P l p O O Variance: P Ph ExPrCX E xeRangelx p tp E XY 2 Vancx Efx P p ! 16

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-p X is called a Bernoulli random variable: X ~ Ber(p) E[X] = E[X 2 ] = p Var(X) = E[X 2 ] – (E[X]) 2 = p – p 2 = p(1-p) Examples: coin flip random binary digit whether a disk drive crashed Jacob (aka James, Jacques) Bernoulli, 1654 – 1705 ! 17