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Probability and StatisticsPart 2. More Probability, Statistics and their Application
Chang-han Rhee
Stanford University
Sep 20, 2011 / CME001
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Outline
StatisticsEstimation ConceptsEstimation Strategies
More ProbabilityExpectation and Conditional ExpectationInterchange of LimitTransforms
SimulationMonte Carlo MethodRare Event Simulation
Further ReferenceClasses at StanfordBooks
2
Outline
StatisticsEstimation ConceptsEstimation Strategies
More ProbabilityExpectation and Conditional ExpectationInterchange of LimitTransforms
SimulationMonte Carlo MethodRare Event Simulation
Further ReferenceClasses at StanfordBooks
3
Probability and Statistics
Probability
Statistics
Model Data
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Estimation
Making best guess of an unknown parameter out of sample data.
eg. Average height of west african giraffe
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Estimator
An estimator (statistic) is a rule of estimation:
θn = g(X1, . . . ,Xn)
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Quality of an Estimator
I BiasEθ − θ
I Variancevar(θ)
I Mean Square Error (MSE)
E[θ − θ]2 = (bias)2 + (var)
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Confidence Interval
Consider the sample mean estimator θ = 1n Sn. From the CLT,
Sn − nEX1√n
D→ σN(0, 1)
Rearranging terms, (note: this is not a rigorous argument)
1n
SnD≈ EX1 +
σ√n
N(0, 1)
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Outline
StatisticsEstimation ConceptsEstimation Strategies
More ProbabilityExpectation and Conditional ExpectationInterchange of LimitTransforms
SimulationMonte Carlo MethodRare Event Simulation
Further ReferenceClasses at StanfordBooks
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Maximum Likelihood Estimation
Finding most likely explanation.
θn = arg maxθ
f (x1, x2, . . . , xn|θ) = f (x1|θ) · f (x2|θ) · f (xn|θ)
I Gold Standard: Gueranteed to beI Often computationally challenging
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Method of Moments
Matching the sample moment and the parametric moments.If θ = (θ1, . . . , θk)∫
xjfθn(x)dx =
1n
n∑i=1
Xji for j = 1, . . . , k
or ∑xjpθn
(x) =1n
n∑i=1
Xji for j = 1, . . . , k
I Statistically, less efficient than MLEI Often computationally efficient
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Outline
StatisticsEstimation ConceptsEstimation Strategies
More ProbabilityExpectation and Conditional ExpectationInterchange of LimitTransforms
SimulationMonte Carlo MethodRare Event Simulation
Further ReferenceClasses at StanfordBooks
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Properties of Expectation
I Jensen’s Inequality
g(EX) ≤ Eg(X) g(·): convex
I Markov’s inequality
P(|X| > x) ≤ E|X|x
x > 0
I Minkovsky’s inequality(E|X + Y|p
)1/p ≤(E|X|p
)1/p+
(E|Y|p
)1/p
I Hölder’s inequality
E|XY| ≤(E|X|p
)1/p(E|Y|p)1/p for 1/p + 1/q = 1
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I If X and Y are independent,
Eg(X)h(Y) = Eg(X)Eh(Y)
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Properties of Conditional Expectation
I Jensen’s Inequality
g(E[X|Y]) ≤ E[g(X)|Y] g(·): convex
I Markov’s inequality
P(|X| > x
∣∣Y) ≤ E[|X|
∣∣Y]x
x > 0
I Minkovsky’s inequality(E[|X + Y|p
∣∣Z])1/p ≤(E[|X|p
∣∣Z])1/p+
(E[|Y|p
∣∣Z])1/p
I Hölder’s inequality
E[|XY|
∣∣Z] ≤ (E[|X|p
∣∣Z])1/p(E[|Y|q
∣∣Z])1/q 1/p + 1/q = 1
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Tower Property
Tower Property (Law of Iterated Expectation, Law of TotalExpectation)
E[X] = E[E[X|Y]
]i.e.,
E[X] =∑x∈S
E[X|Y = y]P(Y = y)
eg.I Y ∼ Unif (0, 1) & X ∼ Unif (Y, 1). What is EX?I Mouse Escape
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Bayes Rule
I The law of total probability:
P(A) =∑
i
P(A|Bi)P(Bi)
I Bayes Rule
P(Ai|B) =P(B|Ai)P(Ai)∑j P(B|Aj)P(Aj)
where A1,A2, . . . ,Ak is a disjoint partition of Ω.
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More Properties of Conditional Expectation
E(Xg(Y)|Y) = g(Y)E(X|Y)E(E(X|Y,Z)|Y) = E(X|Y)
E(X|Y) = X, if X = g(Y) for some g
E(h(X,Y)|Y = y) = Eh(X, y)
E(X|Y) = EX, if X and Y are independent
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Outline
StatisticsEstimation ConceptsEstimation Strategies
More ProbabilityExpectation and Conditional ExpectationInterchange of LimitTransforms
SimulationMonte Carlo MethodRare Event Simulation
Further ReferenceClasses at StanfordBooks
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Monotone Convergence
Theorem (Monotone Convergence)If Xn ≥ 0 and Xn ↑ Xn+1 almost surely, then EXn → EX∞.
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Dominated Convergenceand bounded convergence as a corollary
Theorem (Dominated Convergence)If Xn → X∞ almost surely and |Xn| ≤ Y for all n and some Y such
that EY < ∞, then XnL1→ X∞.
Corollary (Bounded Convergence)If Xn → X∞ almost surely and |Xn| ≤ K for all n and some K ∈ R,
then XnL1→ X∞.
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and more
I Scheffe’s LemmaI Fatou’s LemmaI Uniform IntegrabilityI Fubini’s Theorem
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Outline
StatisticsEstimation ConceptsEstimation Strategies
More ProbabilityExpectation and Conditional ExpectationInterchange of LimitTransforms
SimulationMonte Carlo MethodRare Event Simulation
Further ReferenceClasses at StanfordBooks
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Moment Generating Function and Characteristic Function
Moment generating function and characteristic function chracterizesthe distribution of the random variable.
I Moment Generating Function
MX(θ) = E[exp(θX)]
I Characteristic Function
ΦX(θ) = E[exp(iθX)]
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Outline
StatisticsEstimation ConceptsEstimation Strategies
More ProbabilityExpectation and Conditional ExpectationInterchange of LimitTransforms
SimulationMonte Carlo MethodRare Event Simulation
Further ReferenceClasses at StanfordBooks
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Monte Carlo Method
Computational algorithms that rely on repeated random sampling tocompute their results.
Theoretical BasesI Law of Large Numbers guarantees the convergence
1n(#Xi ∈ A) → P(X1 ∈ A)
I Central Limit Theorem
1n(#Xi ∈ A)− P(X1 ∈ A)
D≈ σ√n
N(0, 1)
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Outline
StatisticsEstimation ConceptsEstimation Strategies
More ProbabilityExpectation and Conditional ExpectationInterchange of LimitTransforms
SimulationMonte Carlo MethodRare Event Simulation
Further ReferenceClasses at StanfordBooks
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Challenges of Rare Event
Probability that a coin lands on its edge.
How many flips do we need to see at least one occurrence?
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Importance Sampling (Change of Measure)
We can express the expectation of a random variable as an expectationof another random variable.
eg.Two continuous random variable X and Y have density fX and fY suchthat fY(s) = 0 implies fX(s) = 0. Then,
Eg(X) =∫
g(s)fX(s)ds =∫
g(s)fX(s)fY(s)
fX(s)ds = Eg(Y)L(Y)
where L(X) = fX(X)fY(X)
.
L(X) is called a likelihood ratio.
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Outline
StatisticsEstimation ConceptsEstimation Strategies
More ProbabilityExpectation and Conditional ExpectationInterchange of LimitTransforms
SimulationMonte Carlo MethodRare Event Simulation
Further ReferenceClasses at StanfordBooks
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Probability
I Basic ProbabilitySTATS 116
I Stochastic ProcessesSTATS 215, 217, 218, 219
I Theory of ProbabilitySTATS 310ABC
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Statistics
I Intro to StatisticsSTATS 200
I Theory of StatisticsSTATS 300ABC
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Application
I Applied StatisticsSTATS 191, 203, 208, 305, 315AB
I Stochastic SystemsMS&E 121, 321
I Stochastic ControlMS&E 322
I Stochastic SimulationMS&E 223, 323, STATS 362
I Little bit of EverythingCME 308
I Econometrics, Finance, Bio and morehttp://explorecourses.stanford.edu
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Outline
StatisticsEstimation ConceptsEstimation Strategies
More ProbabilityExpectation and Conditional ExpectationInterchange of LimitTransforms
SimulationMonte Carlo MethodRare Event Simulation
Further ReferenceClasses at StanfordBooks
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Books
I Sheldon Ross (2009). Introduction to Probability Models.Academic Press; 10th edition
I John A. Rice (2006). Mathematical Statistics and Data Analysis.Duxbury Press; 3rd edition
I Larry Wasserman (2004). All of Statistics : a concise course instatistical inference. Springer, New York
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