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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.
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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.