Random Numbers and Random Numbers and SimulationSimulation

Generating truly random numbers is Generating truly random numbers is not possiblenot possible• Programs have been developed to Programs have been developed to

generate pseudo-random numbersgenerate pseudo-random numbers• Values are generated from Values are generated from

deterministic algorithmsdeterministic algorithms

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© © Fall 2011Fall 2011 John Grego and the University of South CarolinaJohn Grego and the University of South Carolina

Random NumbersRandom Numbers

Pseudo-random deviates can pass any Pseudo-random deviates can pass any statistical test statistical test for randomnessfor randomness

They They appearappear to be independent and to be independent and identically distributedidentically distributed

Random number generators for Random number generators for common distributions are available in common distributions are available in RR

Special techniques (STAT 740) may be Special techniques (STAT 740) may be needed as wellneeded as well

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Monte Carlo SimulationMonte Carlo Simulation Some common uses of simulationSome common uses of simulation• Modeling stochastic behaviorModeling stochastic behavior• Calculating definite integralsCalculating definite integrals• Approximating the sampling Approximating the sampling

distribution of a statistics (e.g., distribution of a statistics (e.g., maximum of a random sample)maximum of a random sample)

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Modeling Stochastic Modeling Stochastic BehaviorBehavior Buffon’s needleBuffon’s needle Random WalkRandom Walk Observe Observe XX11, X, X22, …, , …, where where

p=P(Xp=P(Xii=1)=P(X=1)=P(Xii=-1)=.5 =-1)=.5 and study and study SS11,S,S22,…, ,…, wherewhere

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Si X jj1

i

Modeling Stochastic Modeling Stochastic BehaviorBehavior

This is also called This is also called Gambler’s ruinGambler’s ruin; ; each Xeach Xii represents a $1 bet with a represents a $1 bet with a return of $2 for a win and $0 for a return of $2 for a win and $0 for a loss.loss.

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Gambler’s RuinGambler’s Ruin The properties of a The properties of a fairfair game ( game (p=.5p=.5) )

are a lot more interesting than the are a lot more interesting than the properties of an unfair game (properties of an unfair game (p≠.5p≠.5))

Some properties of this process are Some properties of this process are easy to anticipate (easy to anticipate (E(S)E(S)))

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Gambler’s RuinGambler’s Ruin Some properties are difficult to Some properties are difficult to

anticipate, and can be aided by anticipate, and can be aided by simulation. simulation. • Expected number of returns to 0Expected number of returns to 0• Expected length of a winning streakExpected length of a winning streak• Probability of going broke given an Probability of going broke given an

initial initial bankbank

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Calculating Definite Calculating Definite IntegralsIntegrals

In statistics, we often have to In statistics, we often have to calculate difficult definite integrals calculate difficult definite integrals (posterior distributions, expected (posterior distributions, expected values)values)

(here, (here, x x could be multidimensional)could be multidimensional)

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I h(x)dxa

b

Calculating Definite Calculating Definite IntegralsIntegrals

Example 1Example 1

Example 2Example 2

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I1 4

1 x 2 dx0

1

I2 (4 x12

0

1 2x22 )dx2 dx10

1

Hit-or-Miss Monte CarloHit-or-Miss Monte Carlo

Example 1Example 1

Determine Determine cc such that such that c≥h(x)c≥h(x) across across entire region of interest (here, entire region of interest (here, c=4c=4) )

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h(x) 4

1 x 2

4

1 x 2 dx 4(arctan(1) arctan(0)) 4 /4 0

1

Hit-or-Miss Monte CarloHit-or-Miss Monte Carlo

Generate Generate nn random uniform random uniform (X(Xii,Y,Yii) ) pairs, pairs, XXii’s from ’s from U[a,b]U[a,b] (here, (here, U[0,1]U[0,1]) ) and and YYii’s from ’s from U[0,c] U[0,c] (here, (here, U[0,4]U[0,4]))

Count the number of times (call this Count the number of times (call this mm) that ) that YYii is less than is less than h(Xh(Xii))

Then Then II11 ≈c(b-a)m/n ≈c(b-a)m/n • I.e., (height)(width)(proportion under I.e., (height)(width)(proportion under

curve)curve)

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Classical Monte Carlo Classical Monte Carlo IntegrationIntegration

Take n random uniform values, UTake n random uniform values, U11,…,U,…,Unn over [a,b] and estimate I usingover [a,b] and estimate I using

This method seems straightforward, but This method seems straightforward, but is actually more efficient than Hit-or-Miss is actually more efficient than Hit-or-Miss Monte CarloMonte Carlo

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I h(x)dxa

b

I b a

nh U i

i1

n

Expected Value of a Expected Value of a Function of a Random Function of a Random

VariableVariable Suppose X is a random variable with Suppose X is a random variable with density density ff. Find . Find E[h(x)]E[h(x)] for some for some function function hh, e.g.,, e.g.,

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E X 2 E X

E sin X

Expected Value of a Expected Value of a Function of a Random Function of a Random

VariableVariable

For For nn random values random values XX11, X, X22, …, X, …, Xnn from the distribution of from the distribution of X X (i.e., with (i.e., with density density ff), ),

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n

iiXh

nXhE

1

1

E h X h x X

dx

ExamplesExamples

Example 3: If Example 3: If XX is a random variable is a random variable with a with a N(10,1)N(10,1) distribution, find distribution, find E(XE(X22))

Example 4: If Example 4: If YY is a random variable is a random variable with a with a Beta(5,1)Beta(5,1) distribution, distribution, E(-lnY)E(-lnY)

There are more advanced methods of There are more advanced methods of integration using simulation integration using simulation (Importance Sampling)(Importance Sampling)

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IntegrationIntegration

integrate()integrate() performs numerical performs numerical integration for functions of a integration for functions of a singlesingle variable (variable (notnot using simulation using simulation techniques)techniques)

adapt()adapt() in the in the adaptadapt package package performs multivariate numerical performs multivariate numerical integrationintegration

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Approximating the Sampling Approximating the Sampling Distribution of a StatisticDistribution of a Statistic

To perform inference (CI’s, To perform inference (CI’s, hypothesis tests) based on sampling hypothesis tests) based on sampling statistics, we need to know the statistics, we need to know the sampling distribution of the statistics, sampling distribution of the statistics, at least up to an approximationat least up to an approximation

Example: Example: XX11, X, X22, …, X, …, Xnn ~ iid N(~ iid N(,,22).).

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T X s n

has a t(n 1) distribution

Approximating the Sampling Approximating the Sampling Distribution of a StatisticDistribution of a Statistic

What if the data’s distribution is not What if the data’s distribution is not known?known?• Large sample: Central Limit TheoremLarge sample: Central Limit Theorem• Small sample: Normal theory or Small sample: Normal theory or

nonparametric procedures based on nonparametric procedures based on permutation distributionspermutation distributions

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Approximating the Sampling Approximating the Sampling Distribution of a StatisticDistribution of a Statistic

If the population distribution is known, If the population distribution is known, we can approximate the sampling we can approximate the sampling distribution with simulation.distribution with simulation.• Repeatedly (Repeatedly (mm times) generate random times) generate random

samples of size samples of size nn from the population from the population distributiondistribution

• Calculate a statistic (say, Calculate a statistic (say, SS) each time) each time• The empirical (observed) distribution of The empirical (observed) distribution of S-S-

values approximates the true distribution values approximates the true distribution of of SS

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ExampleExample

XX11, X, X22, X, X33, X, X44 ~Expon(1)~Expon(1) What is the sampling distribution of:What is the sampling distribution of:

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X (the mean)max(X) min(X)

2 (the midrange)