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[ XY ] = 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
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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])
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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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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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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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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 = 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
Probability mass function: Mean: Variance: geometric distribution
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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 = 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-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-p X is called a Bernoulli random variable: X ~ Ber(p) Mean: Variance:
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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-p X is called a Bernoulli random variable: X ~ Ber(p) E[X] = E[X2] = p Var(X) = E[X2] – (E[X])2 = p – p2 = p(1-p) Examples: coin flip random binary digit whether a disk drive crashed
!17 Jacob (aka James, Jacques) Bernoulli, 1654 – 1705