Foundations of Computing II Lecture 20: Continuous Random Variables - PowerPoint PPT Presentation

CSE 312 Foundations of Computing II Lecture 20: Continuous Random Variables Stefano Tessaro tessaro@cs.washington.edu 1

Review – Continuous RVs Probability Density Function (PDF). Cumulative Density Function (CDF). 3 !: ℝ → ℝ s.t. 0 1 = 2 !(%) d% ! % ≥ 0 for all % ∈ ℝ • *+ ,+ ! % d% = 1 • ∫ *+ 0(1) !(%) 1 = 1 Theorem. ! % = 67(8) 68 2

Review – Continuous RVs !(%) 9 : A ℙ < ∈ [9, :] = 2 ! % d% = 0 : − 0(9) @ 3

Uniform Distribution 1 % ∈ [9, :] ! C % = D : − 9 We also say that < < ∼ Unif(9, :) 0 else follows the uniform distribution / is uniformly distributed ,+ 1 1 2 ! C % d% = : − 9 : − 9 = 1 *+ 0 9 : 4

Uniform Density – Expectation 1 % ∈ [9, :] ! C % = D : − 9 < ∼ Unif(9, :) 0 else ,+ M < = 2 ! C % ⋅ % d% *+ A % P A 1 1 = O = : − 9 2 % d% : − 9 2 @ @ : P − 9 P = (: − 9)(9 + :) = 9 + : 1 = 2(: − 9) 2 : − 9 2 5

Uniform Density – Variance 1 % ∈ [9, :] ! C % = D : − 9 < ∼ Unif(9, :) 0 else ,+ M < P = 2 C % ⋅ % P d% ! *+ A A % S 1 1 % P d% = : − 9 2 O = : − 9 3 @ @ = : S − 9 S 3(: − 9) = (: − 9)(: P + 9: + 9 P ) = : P + 9: + 9 P 3(: − 9) 3 6

M < P = : P + 9: + 9 P Uniform Density – Variance M < = 9 + : 2 3 < ∼ Unif(9, :) Var < = M < P − M < P = : P + 9: + 9 P − 9 P + 29: + : P 3 4 = 4: P + 49: + 49 P − 39 P + 69: + 3: P 12 12 = : P − 29: + 9 P = : − 9 P 12 12 7

Exponential Density Assume expected # of occurrences of an event per unit of time is Z • Cars going through intersection • Number of lightning strikes • Requests to web server • Patients admitted to ER Numbers of occurrences of event: Poisson distribution ℙ < = [ = \ *] Z ^ (Discrete) [! How long to wait until next event? Exponential density! Let’s define it and then derive it! 8

Exponential Distribution Definition. An exponential random variable < with parameter Z ≥ 0 is follows the exponential density C % = `Z\ *]8 % ≥ 0 ! 0 % < 0 We write < ∼ Exp Z and say < that follows the exponential distribution. CDF: For 1 ≥ 0 , 3 3 Z\ *]8 d% = Z (−1/Z)\ *]8 = 1 − \ *]3 0 C 1 = 2 c b b 9

[Densities are all 0 on negative Exponential Distribution reals] 2 Z = 2 1.5 Z = 1.5 Z = 1 1 0.5 Z = 0.5 0 0 1 2 3 4 5 6 7 10

Derivation – Number of Cars ] Discretize: In each interval of length 1/j , probability k = l of car arriving. 1/j 1 1 0 1 1 0 1 0 0 0 0 (infinitely many < l = time until first arrival intervals) 3l ℙ < l > 1 = 1 − Z (scaled) geometric j 3l l→+ 1 − Z = \ *]3 = 1 − 0 l→+ ℙ < l > 1 = lim lim C (1) j 11

C % = `Z\ *]8 % ≥ 0 Expectation ! 0 % < 0 ,+ M < = 2 ! C % ⋅ % d% M < = 1 *+ ,+ Z%\ *]8 ⋅ % d% Z = 2 b Var < = 1 o −(% + 1 = 1 Z)\ *]8 = lim p Z P Z o→+ b 12

Memorylessness Definition . A random variable is memoryless if for all q, r > 0 , ℙ < > q + r < > q) = ℙ < > r . Fact . < ∼ Exp(Z) is memoryless. Assuming exp distr, if you’ve waited q minutes, prob of waiting r more is exactly same as q = 0 13

Assuming exp distr, if you’ve waited q minutes, prob of waiting r more is exactly same as q = 0 Memorylessness of Exponential Fact . < ∼ Exp(Z) is memoryless. Proof. ℙ < > q + r < > q) = ℙ < > q + r ∩ < > q ℙ(< > q) = ℙ < > q + r ℙ(< > q) = \ *](t,u) = \ *]u = ℙ(< > r) \ *]t 14

The Normal Distribution Definition. A Gaussian (or normal) random variable with parameters v ∈ ℝ and w ≥ 0 has density Carl Friedrich Pyz \ * {|} ~ Gauss x ! C % = ~~ (We say that < follows the Normal Distribution, and write < ∼ Ä(v, w P ) ) We will see next time why the normal distribution is (in some sense) the most important distribution. 15

Aka a “Bell Curve” (imprecise name) The Normal Distribution 0.4 v = 7 , w P = 1 0.35 0.3 v = 0 , w P = 3 0.25 v = −7 , 0.2 w P = 6 0.15 v = 0 , w P = 8 0.1 0.05 0 -18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18 16

Two Facts Fact. If < ∼ Ä v, w P , then M < = v Proof is easy because density curve is symmetric, and % ⋅ ! C v − % = % ⋅ ! C (v + %) Fact. If < ∼ Ä v, w P , then Var < = w P 17

Shifting and Scaling Fact. If < ∼ Ä v, w P , then É = 9< + : ∼ Ä 9v + :, 9 P w P How do we prove this? (Likely) see next week for a simple proof. Standard (unit) normal = Ä 0, 1 à \ *8 ~ /P d% for Ü ∼ Ä 0, 1 x Definition. Φ Ö = ℙ Ü ≤ Ö = Py ∫ *+ Note: Φ Ö has no closed form – generally given via tables If < ∼ Ä v, w P , then 0 C*â à*â à*â C Ö = ℙ < ≤ Ö = ℙ ≤ = Φ( z ) z z 18

Table of Standard Cumulative Normal Density 19