CPSC 531: Probability Review1 CPSC 531:Probability & Statistics: Review II Instructor: Anirban Mahanti Office: ICT Class Location: TRB 101 Lectures: TR 15:30 – 16:45 hours Class web page: Notes derived from “Probability and Statistics” by M. DeGroot and M. Schervish, Third edition, Addison Wesley, 2002, and “Discrete-event System Simulation” by Banks, Carson, Nelson, and Nicol, Prentice Hall, 2005.

CPSC 531: Probability Review2 Objective and Outline r The world the model-builder sees is probabilistic rather than deterministic. m Some statistical model might well describe the variations. r An appropriate model can be developed by sampling the phenomenon of interest: m Select a known distribution through educated guesses m Make estimate of the parameters m Test for goodness of fit r Goal is to review: m Random variables m Discrete and continuous random variables m Cumulative distribution functions m Expectation, variance, etc.

CPSC 531: Probability Review3 Random Variables r A random variable is a real-valued mapping defined on a sample space. r Suppose that X is a random variable defined on space S, then X assigns a real-number X(s) to each possible outcome s є S. r Typically, X, Y, Z etc denote random variables; x, y, z, etc denote values attained by random variables. r Example: Rolling a pair of dice. Let X be the random variable corresponding to the sum of the dice on a roll. If we think of the sample points as a pair (i, j), where i = value rolled by the first dice and j = value rolled by the second dice, we have: X(s) = i+j

CPSC 531: Probability Review4 Discrete Random Variables r A random variable X is said to be discrete if the number of possible values of X is finite, or at most, an infinite sequence of different values. r Example: Consider jobs arriving at a job shop. Let X be the number of jobs arriving each week at a job shop. S = possible values of X (range space of X) = {0,1,2,…} p(x i ) = probability the random variable is x i = P(X = x i ) m p(x i ), i = 1,2, … must satisfy: m The collection of pairs [x i, p(x i )], i = 1,2,…, is called the probability distribution of X, and p(x i ) is called the probability mass function (pmf) of X. m The pmf is referred to as “probability function” in some texts

CPSC 531: Probability Review5 Discrete Random Variables r Consider a random variable X that takes on values 1, 2, 3, and 4 with probabilities 1/6, 1/3, 1/3, and 1/6, resp. 12 p(x)p(x) x

CPSC 531: Probability Review6 Continuous Random Variables r X is a continuous random variable if there exists a non-negative function f(x) such that for any set of real numbers A є S r The probability that X lies in the interval [a,b] is given by: r f(x), denoted as the pdf of X, satisfies: r Properties

CPSC 531: Probability Review7 Continuous Random Variables r Example: Life of an inspection device is given by X, a continuous random variable with pdf: m X has an exponential distribution with mean 2 years m Probability that the device’s life is between 2 and 3 years is:

CPSC 531: Probability Review8 Cumulative Distribution Function r The cumulative distribution function (cdf) of a random variable X is a function F(x), defined for each real number x: m F(x) = P(X <= x) for -∞ < x < ∞ m If X is discrete, then m If X is continuous, then r Properties r All probability question about X can be answered in terms of the cdf, e.g.:

CPSC 531: Probability Review9 Cumulative Distribution Function r Example: An inspection device has cdf: m The probability that the device lasts for less than 2 years: m The probability that it lasts between 2 and 3 years:

CPSC 531: Probability Review10 Expectation r The expected value of X is denoted by E(X) m If X is discrete m If X is continuous m The mean, μ, is the 1 st moment of X m A measure of the central tendency r Properties: m E(cX) = cE(X), where c is a constant m E(Y) = aE(X) + b, where Y=aX+b, a & b are constants m E(X + Y) = E(X) + E(Y) regardless of whether X and Y are independent m E(X.Y) = E(X).E(Y) if X & Y are independent

CPSC 531: Probability Review11 Variance  The variance of X is denoted by V(X) or var(X) or  2 m Definition: V(X) = E[(X – E[X] 2 ] m Also, V(X) = E(X 2 ) – [E(x)] 2 m The variance is a measure of the dispersion or spread of a random variable about its mean  The standard deviation of X is denoted by  m Definition: square root of V(X) m Expressed in the same units as the mean r Properties: m V(cX) = c 2 V(X) m V(X + Y) = V(X) + V(Y) if X, Y are independent

CPSC 531: Probability Review12 Density functions for continuous random variables with large and small variances (Source LK00, Fig 4.6) µ µ σ 2 large σ 2 small X X X X Small vs. Large Variance

CPSC 531: Probability Review13 Expectations and Variance (example) r Example: The mean of life of the previous inspection device is: r To compute variance of X, we first compute E(X 2 ): r Hence, the variance and standard deviation of the device’s life are:

CPSC 531: Probability Review14 Joint Distributions r Let X and Y each have a discrete distribution. Then X and Y have a discrete joint distribution if there exists a function p(x,y) such that: p(x,y) = P[X=x and Y=y] r Random variables X and Y are jointly continuous if there exists a non-negative function f(x,y) called the joint probability density function of X and Y, such that for all sets of real numbers A and B P(X є A, Y є B) = ∫ ∫f(x,y)dxdy BA

CPSC 531: Probability Review15 Covariance r 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) r The covariance is a measure of the dependence between X and Y. Note that Cov(X, X) = V(X).

CPSC 531: Probability Review16 Covariance Cov(X, Y) X and Y are = 0uncorrelated > 0positively correlated < 0negatively correlated Independent random variables are also uncorrelated.

CPSC 531: Probability Review17 Statistical Models r Application areas where statistical models find widespread use: m Queueing systems m Inventory and supply-chain systems m Reliability and maintainability m Limited data

CPSC 531: Probability Review18 Queueing Systems r In a queueing system, interarrival and service-time patterns can be probabilistic (e.g., our M/M/1 example). r Sample statistical models for interarrival or service time distribution: m Exponential distribution: if service times are completely random m Normal distribution: fairly constant but with some random variability (either positive or negative) m Truncated normal distribution: similar to normal distribution but with restricted value. m Gamma and Weibull distribution: more general than exponential (involving location of the modes of pdf’s and the shapes of tails.)

CPSC 531: Probability Review19 Inventory and supply chain r In realistic inventory and supply-chain systems, there are at least three random variables: m The number of units demanded per order or per time period m The time between demands m The lead time r Sample statistical models for lead time distribution: m Gamma r Sample statistical models for demand distribution: m Poisson: simple and extensively tabulated. m Negative binomial distribution: longer tail than Poisson (more large demands). m Geometric: special case of negative binomial given at least one demand has occurred.

CPSC 531: Probability Review20 Reliability and maintainability r Time to failure (TTF) m Exponential: failures are random m Gamma: for standby redundancy where each component has an exponential TTF m Weibull: failure is due to the most serious of a large number of defects in a system of components m Normal: failures are due to wear

CPSC 531: Probability Review21 Our next stop r Discrete distributions, such as: m Bernoulli trials and Bernoulli distribution m Binomial distribution m Geometric and negative binomial distribution m Poisson distribution r Continuous distributions, such as: m Uniform m Exponential m Normal m Weibull m Lognormal