2. Review of Probability and Statistics

2. Review of Probability and Statistics Refs: Law & Kelton, Chapter 4

Random Variables • Experiment: a process whose outcome is not known with certainty • Sample space: S = {all possible outcomes of an experiment} • Sample point: an outcome (a member of sample space S) • Example: in coin flipping, S={Head,Tail}, Head and Tail are outcomes • Random variable: a function that assigns a real number to each point in sample space S • Example: in flipping two coins: • If X is the random variable = number of heads that occur • then X=1 for outcomes (H,T) and (T,H), and X=2 for (H,H)

Random Variables: Notation • Denote random variables by upper case letters: X, Y, Z, ... • Denote values of random variables by lower case letters: x, y, z, … • The distribution function (cumulative distribution function) F(x) of random variable X for real number x is • F(x) = P(Xx) for -<x< • where P(Xx) is the probability associated with event {Xx} • F(x) has the following properties: • 0F(x) 1 for all x • F(x) is nondecreasing: i.e. if x1x2 then F(x1)  F(x2)

Discrete Random Variables • A random variable is discrete if it takes on at most a countable number of values • The probability that random variable X takes on value xi is given by: • p(xi) = P(X=xi) for I=1,2,… • and • p(x) is the probability mass function of discrete random variable X • F(x) is the probability distribution function of discrete random variable X

Continuous Random Variables X is a continuous random variable if there exists a non-negative function f(x) such that for any set of real numbers B, f(x) is the probability density function for the continuous RV X: F(x) is the probability distribution function for the continuous RV X:

Distribution and density functions of a Continuous Random Variable Given an interval I = [a,b] f(x) a b x

Joint Random Variables In the M/M/1 queuing system, the input can be represented as two sets of random variables: arrival times of customers: A1, A2, …, An and service times of customers:S1, S2, …, Sn The output can be a set of random variables: delays in queue of customers: D1, D2, …, Dn The D’s are not independent.

Jointly discrete Random Variables Consider the case of two discrete random variables X and Y, the joint probability mass function is p(x,y) = P(X=x,Y=y) for all x,y X and Y are independent if p(x,y) = pX(x)pY(y) for all x,y these are the marginal probability mass functions

Jointly continuous Random Variables Consider the case of two continuous random variables X and Y. X and Y are jointly continuous random variables if there exists a non-negative function f(x,y) (the joint probability density function of X and Y) such that for all sets of real numbers A and B, X and Y are independent if fX(x) and fY(y) are called the marginal probability density functions

Measuring Random Variables: mean and median Consider n random variables X1, X2, …, Xn The mean of the random variable Xi (i=1,2,…,n) is The mean is the expected value, and is a measure of central tendency. The median, x0.5 , is the smallest value of x such that FX(x)  0.5 For a continuous random variable, F(x0.5) = 0.5

Measuring Random Variables: variance The variance of the random variable Xi , denoted by i2 or Var(Xi) is i2 = E[(Xi-i)2] = E(Xi2) - i2 (this is a measure of dispersion) small 2 large 2   Some properties of variance are: Var(X)  0 Var(cX) = c2 Var(X) If the Xi’s are independent, then Var(X + Y) = Var(X) + Var(Y) or Standard deviation is

Variance is not dimensionless e.g. if we collect data on service times at a bank machine with mean 0.5 minutes, variance 0.1 minute, the same data in seconds will give mean 30 seconds, variance 360 seconds; i.e., variance is dependent on scale! *602 *60

Measures of dependence The covariance between random variables Xi and Xj where i=1,…,n and j=1,…,n is Cij = Cov(Xi , Xj) = E[(Xi - i)(Xj- j)] = E[Xi Xj] - i j Some properties of covariance: Cij = Cji if i=j, then Cij = Cji = Cii = Cjj = i2 if Cij > 0, then Xi and Xj are positively correlated: Xi > i and Xj > j tend to occur together, and Xi < i and Xj < j tend to occur together if Cij < 0, then Xi and Xj are negatively correlated: Xi > i and Xj < j tend to occur together, and Xi < i and Xj > j tend to occur together Cij is NOT dimensionless, i.e. the value is influenced by the scale of the data.

A dimensionless measure of dependence Correlation is a dimensionless measure of dependence: Var(a1X1+ a2X2 ) = a12Var(X1)+2a1a2Covar(X1, X2)+ a22Var(X1) Var(X-Y) = Var(X) + Var(Y) – 2Cov(X,Y) Var(X+Y) = Var(X) + Var(Y) +2Cov(X,Y)

The covariance between the random variables X and Y, denoted by Cov(X, Y), is defined by Cov(X, Y) = E{[X - E(X)][Y - E(Y)]} = E(XY) - E(X)E(Y) The covariance is a measure of the dependence between X and Y. Note that Cov(X, X) = Var(X).

Definitions: Cov(X, Y) X and Y are = 0 uncorrelated > 0 positively correlated < 0 negatively correlated Independent random variables are also uncorrelated.

Note that, in general, we have Var(X - Y) = Var(X) + Var(Y) - 2Cov(X, Y) If X and Y are independent, then Var(X - Y) = Var(X) + Var(Y) The correlation between the random variables X and Y, denoted by Cor(X, Y), is defined by

It can be shown that -1  Cor(X, Y)  1

4.2. Simulation Output Data and Stochastic Processes A stochastic process is a collection of "similar" random variables ordered over time all defined relative to the same experiment. If the collection is X1, X2, ... , then we have a discrete-time stochastic process. If the collection is {X(t), t 0}, then we have a continuous-time stochastic process.

Example 4.3: Consider the single-server queueing system of Chapter 1 with independent interarrival times A1, A2, ... and independent processing times P1, P2, ... . Relative to the experiment of generating the Ai's and Pi's, one can define the discrete-time stochastic process of delays in queue D1, D2, ... as follows: D 1 = 0 Di +1 = max{Di + Pi - Ai +1, 0} for i = 1, 2, ...

Thus, the simulation maps the input random variables into the output process of interest. Other examples of stochastic processes: • N1, N2, ... , where Ni = number of parts produced in the ith hour for a manufacturing system • T1, T2, ... , where Ti = time in system of the ith part for a manufacturing system

• {Q(t), t 0}, where Q(t) = number of customers in queue at time t •C1, C2, ... , where Ci = total cost in the ith month for an inventory system •E1, E2, ... , where Ei = end-to-end delay of ith message to reach its destination in a communications network • {R(t), t 0}, where R(t) = number of red tanks in a battle at time t

Example 4.4: Consider the delay-in-queue process D1, D2, ... for the M/M/1 queue with utilization factor ρ. Then the correlation function ρj between Diand Di+j is given in Figure 4.8.

ρj ρ = 0.9 1.0 0.9 0.8 0.7 0.6 0.5 ρ = 0.5 0.4 0.3 0.2 0.1 j 0 1 2 3 4 5 6 9 10 7 8 Figure 4.8. Correlation function ρj of the process D1, D2, ... for the M/M/1 queue.

Estimating distribution parameters Assume that X1, X2, …, Xn are independent, identically distributed (IID) random variables with finite population mean , and finite population variance 2. We would like to estimate  and 2 The sample mean is This is an unbiased estimator of , i.e. also, for a very large number of independent calculations of , the average of the will tend to 

Estimating variance The sample variance is This is an unbiased estimator of , i.e. E[s2(n)] = 2 this is a measure of how well estimates 

Two interesting theorems Central Limit Theorem: For large enough n, tends to be normally distributed with mean  and variance 2/n This means that we can assume that the average of a large numbers of random samples is normally distributed - very convenient. Strong Law of Large Numbers: For an infinite number of experiments, each resulting in , for sufficiently large n, will be arbitrarily close to  for almost all experiments. This means that we must choose n carefully: every sample must be large enough to give a good .

Area = 0.68 Figure 4.7. Density function for a N(, 2)distribution.

Confidence Intervals f(x) 1- x 0 Interpretation of confidence intervals: If one constructs a very large number of independent 100(1-)% confidence intervals, each based on n observations, with n sufficiently large, then the proportion of these confidence intervals that contain  should be 1-. What is n sufficiently large? The more “non-normal” the distribution of Xi , the larger n must be to get good coverage (1- ).

An exact confidence interval If the Xi’s are normal random variables, then tn has a t distribution with n-1 degrees of freedom (df), and an exact 100(1-)% confidence interval for  is: This is better than z’s confidence interval, because it provides better coverage (1- ) for small n, and converges to the z confidence interval for large n. Note: this estimator assumes that the Xi’s are normally distributed. This is reasonable if the central limit theorem applies: i.e. each Xi is the average of one sample (large enough to give a good average), and we take a large number of samples (enough samples to get a good confidence interval).

2. Review of Probability and Statistics