Ch.4 Review of Basic Probability and Statistics

4.1 Introduction

Probability and

statistics

Model a probabilistic

system

Model a probabilistic

system

Validate the simulation

model

Validate the simulation

model

Choose the input probabilistic

distribution

Choose the input probabilistic

distribution

Generate random samples from the input distribution

Generate random samples from the input distribution

Perform statistic analyses of the

simulation output data

Perform statistic analyses of the

simulation output data

Design the simulation

experiments

Design the simulation

experiments

4.2 Random variables and their properties

• Experiment is a process whose outcome is not known with certainty.

• Sample space (S) is the set of all possible outcome of an experiment.

• Sample points are the outcomes themselves.• Random variable (X, Y, Z) is a function that

assigns a real number to each point in the sample space S.

• Values x, y, z

Examples

• flipping a coin S={H, T}• tossing a die S={1,2,…,6}• flipping two coins S={(H,H), (H,T), (T,H), (T,T)} X: the number of heads that occurs• rolling a pair of dice S={(1,1), (1,2), …, (6,6)} X: the sum of the two dice

Distribution (cumulative) function

)( xXP

xxXPxF for )()(

}{ xX : the probability associated with the event

Properties:

1.

2. F(x) is nondecreasing [i.e., if ].

3. .

. allfor 1)(0 xxF

)()( then, 2121 xFxFxx

0)(lim and 1)(limx

xFxFx

Discrete random variableA random variable X is said to be discrete if it can take on at most a

countable number of values.

The probability that X takes on the value

Then

Probability mass function on I=[a,b]

1x

,...2,1for )()( ixXPxp ii

1

1)(i

ixp

xxpxF

xpIXP

xxi

bxai

i

i

allfor )()(

)()(

Examples

0 1 2 3 4 x

p(x)1

1/6

1/3

1/2

2/3

5/6

p(x) for the demand-size random variable X.

32

31

31)3()2()32( ppXP

0 1 2 3 4 x

F(x)1

1/6

1/3

1/2

2/3

5/6

F(x) for the demand-size random variable X.

Continuous random variables

PX B B

fxdx

A random variable is said to be continuous if there exists a nonnegative function f(x) such that for any set of real number B

and

fxdx 1

f(x) is called the probability density function.

PX x PX x, x x

xfydy 0

PX x, x x x

x xfydy

Fx PX , x

xfydy for all x

)()( xFxf

PX I a

bfydy Fb Fa I a, b

x

f(x)

xx xx ''xx

]),[( xxxXP

]),[( '' xxxXP

Interpretation of the probability density function

Uniform random variable on the interval [0,1]

fx 1 if 0 x 1

0 otherwise

0 x 1If

xdydyyfxF xx 1)()( 00

, then

0 x

f(x)

1

1

f(x) for a uniform random variable on [0,1]

fx 1 if 0 x 1

0 otherwise

0 x

F(x)

1

1

F(x) for a uniform random variable on [0,1]

PX x, x x x

x xfydy

Fx x Fx x x x

x

0 x x x 1where

Exponential random variable

0 x

f(x)

1

0 x

F(x)

1

f(x) for an exponential random variable with mean

F(x) for an exponential random variable with mean

Joint probability mass function

If X and Y are discrete random variables, then let

px, y PX x, Y y for all x, y

where p(x,y) is called the joint probability mass function of X and Y.

X and Y are independent if

px, y pXxpYy for all x, y

where

pXx all y

px, y

pYy all x

px, y

are the (marginal) probability mass functions of X and Y.

Example 4.9

Suppose that X and Y are jointly discrete random variables with

px, y xy27 for x 1, 2 and y 2, 3, 4

0 otherwise

Then

pXx y 2

4xy27 x

3

pYy x 1

2xy27 y

9

for x=1,2

for y=2,3,4

Since px, y xy/27 pXxpYy For all x, y, the random variables X and Y

are independent.

Joint probability density function

The random variables X and Y are jointly continuous if there exists a nonnegative function f(x,y), such that for all sets of real numbers A and B,

PX A, Y B B

Afx, ydxdy

X and Y are independent if

fx, y fXxfYy for all x and y

where

fXx

fx, ydy

fYy

fx, ydx

are the (marginal) probability density functions of X and Y, respectively.

Example 4.11Suppose that X and Y are jointly continuous random variables with

fx, y 24xy for x 0, y 0, and x y 1

0 otherwise

Then

fXx 0

1 x24xydy 12xy2|0

1 x 12x1 x 2 for 0 x 1

fYy 0

1 y24xydx 12yx2

01 y 12y1 y 2 0 y 1for

Since

f 12 , 1

2 6 32 2 fX 1

2 fY 12

X and Y are not independent.

Mean or expected value

Ex i i j 1

x jpX i x j if X i is discrete

xfX i x if X i is continuous

The mean is one measure of central tendency in the sense that it is the center of gravity

Examples 4.12-4.13For the demand-size random variable, the mean is given by

1 16

2 13

3 13

4 16

52

For the uniform random variable, the mean is given by

0

1xfxdx

0

1xdx 1

2

Properties of means

EcX cEX

Ei 1n c iX i

i 1n c iEX i

X i

1.

Even if the ‘s are dependent.

2.

Median

The median of the random variable is defined to be thesmallest value of x such that

x0.5

x0.5FX i x0.5 0. 5

x0.5 x

)(xfiX

area=0.5

The median for a continuous random variable x0.5

Example 4.14

The median may be a better measure of central tendency than the mean.

1. Consider a discrete random variable X that takes on each of the values, 1, 2, 3, 4, and 5 with probability 0.2. Clearly, the mean And the median of X are 3.

2. Now consider random variable Y that takes on each of the values, 1, 2, 3, 4, and 100 with probability 0.2. The mean and the median of X are 22 and 3, respectively.

Note that the median is insensitive to this change in the distribution.

Variance

])[()( 22iiii XEXVar 2222 )]([)()( iiii XEXEXE

For the demand-size random variable,

EX2 12 16

22 13

32 13

42 16

436

VarX EX2 2 436

52

2 1112

For the uniform random variable on [0,1],

EX2 0

1x2fxdx

0

1x2dx 1

3

VarX EX2 2 13

12

2 112

2large

2small

Density functions for continuous random variables with large and small variances.

Properties of the variance

VarX 0

VarcX c2VarX

Vari 1n X i

i 1n VarX i

1.

2.

3. X iif the ‘s are

independent (or uncorrelated).

Standard deviation

ii 96.1

i i2

iX ii 96.1The probability that is between and is 0.95.

Covariance

jijijjiiijji XXEXXECXXCov )()])([(),(

The covariance between the random variables and is a measure of their dependence.

X i X j

C ij C ji

C ij C ji i2 if i=j,

Example 4.17

For the jointly continuous random variables X and Y in Example 4.11

EXY 0

1 0

1 xxyfx, ydydx

0

1x2

0

1 x24y2dydx

0

18x21 x 3dx

215

EX 0

1xfXxdx

0

112x21 x 2dx 2

5

EY 0

1yf Yydy

0

112y21 y 2dy 2

5CovX, Y EXY EXEY

215

25 2

5

275

If and are independent random variables

C ij 0

X i X j

X jX i and are uncorrelated.

Generally, the converse is not true.

Correlated

If , then and are said to be positively correlated.X i X jC ij 0

If , then and are said to be negatively correlated.X i X j0ijC

X i i

X j j

X i i

X j j

and tend to occur together

and tend to occur together

iiX

X j j

iiX X j j

and tend to occur together

and tend to occur together

Correlation

ij C ij

i2 i

2

i 1, 2, , n

j 1, 2, , n

1 ij 1

ijIf is close to +1, then and are highly positively correlated. X i X j

ijIf is close to -1, then and are highly negatively correlated. X i X j

For the random variable in Example 4.11

VarX VarY 125

CorX, Y CovX, YVarXVarY

2

75125

23

4.3 Simulation output data and stochastic processes

State space is the set of all possible values that these random variables can take on.

X1,X2,

Continuous-time stochastic process: Xt, t 0

Stochastic process is a collection of "similar" random variables ordered over time, which are all defined on a common sample space.

Discrete-time stochastic process:

Example 4.19 M/M/1 queue with A1,A2,

IID service times S1,S2, FIFO service

D1,D2,

D1 0

Di 1 maxDi Si Ai 1, 0 for i 1,2,

Di Di 1 are positively correlated.

IID interarrival times

Define the discrete-time stochastic process of delays in queue

and

input random variables output stochastic processsimulation

The state space: the set of nonnegative real numbers

Example 4.20

For the queueing system of Example 4.19,

Let be the number of customers in the queue at time t .

Then is a continuous-time stochastic process with state space

Qt

Qt, t 00,1,2,

Covariance-stationary

Assumptions about the stochastic process are necessary to draw inferences in practice.

A discrete-time stochastic process X1,X2, is said to be

i for i 1,2, and i

2 2 for i 1,2, and 2

covariance-stationary, if

and is independent of i for ),(, jiijii XXCovC j 1,2, .

Covariance-stationary process

For a covariance-stationary process, the mean and variance are stationary over time, and the covariance between two observations and depends only on the separation j and not actual time value i and i+j.

Xi Xi j

We denote the covariance and correlation between and byXi Xi j

Cj j

,2,1for 0

222

,

jC

CCC

jjj

jii

jii

and respectively, where

Example 4.22

Consider the output process for a covariance-stationary M/M/1 queue with .

D1,D2, / 1

Warmup period

In general, output processes for queueing systems are positively correlated.

If is a stochastic process beginning at time 0 in a simulation, then it is quite likely not to be covariance-stationary.

However, for some simulation will be approximately covariance-stationary if k is large enough, where k is the length of the warmup period.

X1,X2,

Xk 1,Xk 2,

4.4 Estimation of means, variance, and correlations

Suppose are IID random variables with finite population mean and finite population variance

nXXX ,,2,1 2

)]([ nXESample mean Xn

i 1

n

Xi

n

Sample variance 2 22 )]([ nSE S2n i 1

n

Xi Xn2

n 1

Unbiased estimators:

X X

Density function for )(nX

First observationof

Second observationof )(nX )(nX

How close is to ? nX

VarXn Var 1n

i 1

n

Xi

1n2

Vari 1

n

Xi (because the Xi’s are independent)

1n2

i 1

n

VarXi

1n2

n 2 2

n

VarXn S2nn

i 1

n

Xi Xn2

nn 1

How close is to nX to construct a confidence interval

Unbiased estimator

n small

Distributions of for small and large n.

n large

Density function

for X n

Density function

for X n

X n

Estimate the variance of the sample mean .

iX

Var X n

´s are independent iX ´s are uncorrelated 0j

However, the simulation output data are almost always correlated.

nXXX ,,, 21 are from a covariance-stationary stochastic process,

Then, is an unbiased estimator of , however, is no longer an unbiased estimator of . Since

ES2n 2 1 2

j 1

n 1

1 j/n j

n 1

X n S2n 2

However, simulation output data are always correlated. Since

ES2n 2 1 2

j 1

n 1

1 j/n j

n 1

j 0 22 ))(( nSE

For a covariance-stationary process:

Var X n 2

1 2 j 1

n 1

1 j/n j

n (2)

(1)

If one estimates from (correct in the IID case)

Var X n S2n/n

there are two errors: • the bias in as an estimator of .• the negligence of the correlation terms in Eq. (2).

S2n 2

Solution: combine Eq. (1) and Eq. (2)

E S2nn n/an 1

n 1Var X n

an 1 2 j 1

n 1

1 j/n j

j 0If , then and .an 1 ES2n/n Var X n

(3)

Example 4.24

D1, D2, , D10 from the process of delays for a covariance-stationary M/M/1 queue with . Eq.(1) and (3)

ES210 0. 0328 2

E S21010

0. 0034VarD 10

2 VarD i D 10 i 1

10

D i

10S210

i 1

10

D i D 102

9

Thus, is a gross underestimate of , and we are likely to be overly optimistic about the closeness of to

S210/10 VarD 10D 10 ED i

0. 9

Estimate . j

j Ĉ j

S2n, Ĉ j

i 1

n j

X i X nX i j X n

n j

In general "good" estimates of the 's will be difficult to obtain unless n is very large and j is small relative to n.

j

4.5.1 Confidence Intervals

nnXZn //])([ 2

)()( zZPzF nn

4.5.1 Confidence Intervals

Central Limit Theorem: as , where , the distribution function of a normal random variable with and , is given by

If n is "sufficiently large", the random variable will be

approximately distributed as a standard normal random variable, regardless of the underlying distribution of the 's. For large n, the sample mean is approximately distributed as a normal random variable with mean and variance

Fnz z n z

0 2 1

z 12

ze y2/2dy for z

Zn

Xi

Xn 2/n.

tn Xn / S2n/n

nnXZn //])([ 2

P z1 /2 Xn S2n/n

z1 /2

PXn z1 /2S2n

n

Xn z1 /2S2n

n

1

where ( ) is the upper critical point for a standard normal random variable.

z1 /2 0 1 1 /2

f(x)

Shaded area = 1

2/1 z 2/1 z0 x

Xn z1 /2S2n

n confidence interval

If n is sufficiently large, an approximate percent confidence interval for is given by

1001

Interpretation I: If one constructs a very large number of independent percent confidence intervals, each based on n observations, where n is sufficiently large, the proportion of these confidence intervals that contains (cover) should be .

1001

1

Interpretation II: If the 's are normal random variables, the random variable has a t distribution with n-1 degree of freedom (df), and an exact (for any ) percent confidence interval for is given by

Where is the upper critical point for the t distribution with n-1 df

tn Xn / S2n/nn 2

1001

Xn tn 1,1 /2S2n

n

tn 1,1 /2 1 /2

iX

f(x)

x

Standard normal distributiont distribution with 4df

0

Figure 4.16 Density function for the t distribution with 4df and for the standard normal distribution.

Example 4.26

10 observations: 1.20, 1.50, 1.68, 1.89, 0.95, 1.49, 1.58, 1.55,

0.50, 1.09 are from a normal distribution,

To construct a 90% confidence interval for .

X10 t9,0.95S210

10 1.34 1.83 0.17

10 1.34 0.24

X10 1.34 S210 0.17

v EX 3 23/2

v

Distribution Skewness v n=5 n=10 n=20 n=40

Normal 0.00 0.910 0.902 0.898 0.900

Exponential 2.00 0.854 0.878 0.870 0.890

Chi Square 2.83 0.810 0.830 0.848 0.890

Lognormal 6.18 0.758 0.768 0.842 0.852

Hyperexponential 6.43 0.584 0.586 0.682 0.774

Table 4.1 Estimated coverages based on 500 experiments

4.5.2 Hypothesis tests for the mean

H0

0

H0

:

If is large, is not likely to be true.|Xn 0 |

If is true, the statistic will have a t distribution with n-1 df.

tn Xn 0/ S2n/n

H0

If |tn | tn 1,1 /2

tn 1,1 /2

reject H0

"accept" H0

H0

Example 4.27

To test the null hypothesis that at level .H0 1 0.10

For Example 4.26,

t10 X10 1

S210/10 0.34

0.17/10 2.65 1.83 t9,0.95

We reject . H0

4.6 The Strong Law of Large Numbers

Theorem 4.2 Xn w.p. 1 as n .

Example 4.29

Tarea II: Teoría de la probabilitad y estatística

A.M. Law and W.D. Kelton, Simulation, Modeling and Analysis, 3rd edition, pp. 261-263.

Problems 4.1, 4.2, 4.4, 4.7, 4.9, 4.10, 4.13, 4.20, 4.21, 4.23, 4.24, 4.25, 4.26