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Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: 1. Sums of random variables 2. Farha/Antonius 3. Continuous Random Variables, Density, Uniform, Normal 4. LLN & CLT. u u 1) E(X+Y) = E(X) + E(Y). Whether X & Y are independent or not! Similarly, E(X + Y + Z + …) = E(X) + E(Y) + E(Z) + … And, if X & Y are independent, then V(X+Y) = V(X) + V(Y). so SD(X+Y) = √[SD(X)^2 + SD(Y)^2]. Also, if Y = 9X, then E(Y) = 9E(Y), and SD(Y) = 9SD(X). V(Y) = 81V(X). 2) Farha vs. Antonius. Running it 4 times. Let X = chips you have after the hand. Let p be the prob. you win. X = X1 + X2 + X3 + X4, where X1 = chips won from the first “run”, etc. E(X) = E(X1) + E(X2) + E(X3) + E(X4) = 1/4 pot (p) + 1/4 pot (p) + 1/4 pot (p) + 1/4 pot (p) = pot (p) = same as E(Y), where Y = chips you have after the hand if you ran it once!!! But the SD is smaller: clearly X1 = Y/4, so SD(X1) = SD(Y)/4. So, V(X1) = V(Y)/16. V(X) ~ V(X1) + V(X2) + V(X3) + V(X4), = 4 V(X1) = 4 V(Y) / 16 = V(Y) / 4. So SD(X) = SD(Y) / 2. 3) Continuous Random Variables, Density, Uniform, Normal Density (or pdf = Probability Density Function) f(y): ∫B f(y) dy = P(X in B). Expected value (µ) = ∫ y f(y) dy. (= ∑ y P(y) for discrete X.) Example 1: Uniform (0,1). f(y) = 1, for y in (0,1). µ = 0.5. s = 0.29. P(X is between 0.4 and 0.6) = ∫.4 .6 f(y) dy = ∫.4 .6 1 dy = 0.2. Example 2: Normal. mean = µ, SD = s, 68% of the values are within 1 SD of µ 95% are within 2 SDs of µ Example 3: Standard Normal. Normal with µ = 0, s = 1. 95% between -1.96 and 1.96 4) Law of Large Numbers, CLT Sample mean (X) = ∑Xi / n iid: independent and identically distributed. Suppose X1, X2 , etc. are iid with expected value µ and sd s , LAW OF LARGE NUMBERS (LLN): X ---> µ . CENTRAL LIMIT THEOREM (CLT): (X - µ) ÷ (s/√n) ---> Standard Normal. Useful for tracking results. Note: LLN does not mean that short-term luck will change. Rather, that short-term results will eventually become negligible. 95% between -1.96 and 1.96 Truth: -49 or 51, each with prob. 1/2. exp. value = 1.0 Truth: -49 to 51, exp. value = 1.0 Estimated as X +/- 1.96 s/√n = .95 +/- 0.28 * Poker has high standard deviation. Important to keep track of results. * Don’t just track ∑Xi. Track X +/- 1.96 s/√n . Make sure it’s converging to something positive.