Operations Management Session 10: Probability Concepts

Operations Management Simulation Game Game codes due. Please go to http://usc.responsive.net/lt/usc/start.html to register. Course code: usc. Individual code: what you purchased from the bookstore. Case groups posted. Please double-check. Session 10 Operations Management

Operations Management Today’s Class Probability Concept Review Basic Statistics Formula Common Distribution Session 10 Operations Management

Operations Management Quote of the day Without the element of uncertainty, the bringing off of even, the greatest business triumph would be dull, routine, and eminently unsatisfying. J. Paul Getty Session 10 Operations Management

Operations Management Blackjack You have a 9 and 5, what will happen if you hit? Session 10 Operations Management

Operations Management Random Experiment Random Experiment: An experiment in which the precise outcome is not known ahead of time. The set of possibilities however is known Examples: Demand for blue blazers next month The value of a rolled die The waiting times of customers in the bank The waiting time for an ATT service person Tomorrow’s closing value of the NASDAQ The temperature in Los Angeles tomorrow Session 10 Operations Management

Operations Management Random Variable A random variable is the numerical value determined by the outcome of a random experiment A random variable can be discrete (i.e. takes on only a finite set of values) or continuous Examples: The value on a rolled die is a discrete random variable The demand for blazers is a discrete random variable The birth weight of a newborn baby is a continuous variable The waiting time for the AT&T service person is a continuous random variable Session 10 Operations Management

Operations Management Sample Space Sample space is the list of possible outcomes of an experiment Examples: For a die, the sample space S is: {1,2,3,4,5,6} For the demand for blue blazers it is all possible realizations of the demand. For example: {1000,1001,1002…,2000} The waiting time in the bank is any number greater than or equal to 0. This is a continuous random variable The waiting time for a bus at a bus stop is any number between 0 and 30 minutes. This is a continuous random variable that is bounded Session 10 Operations Management

Operations Management Event An event is a set of one or more outcomes of a random experiment Examples: Getting less than 5 by rolling the die: This event occurs if the values observed are {1,2,3, or 4} The demand is smaller or equal to 1500. This event occurs if the values of the demand are {1000, 1001, … 1500} The waiting time for a bus at the bus stop exceeds 10. This event occurs if the wait time is in the interval (10, 30) Session 10 Operations Management

Operations Management Probability The probability of an event is a number between 0 and 1 1 means that the event will always happen 0 means that the event will never happen The probability of an event A is denoted as either P(A) or Prob(A) Example: Probability of Rolling die and observing a number less than 5 = P(outcome< 5) = Prob(observing {1,2,3 or 4}) = 4/6 = 2/3 Session 10 Operations Management

Operations Management Probability Probability that A doesn’t occur: P(not A) = 1 – P(A) Thus, the probability you will roll a number larger or equal to 5 is or 6 is: 1 – Probability (Outcome <5) = 1 – 2/3 = 1/3 Session 10 Operations Management

Operations Management Probability Suppose all the outcomes that constitute the “waiting time” for an AT&T operator are equally likely. The minimum waiting time is 30 min and the maximum is 90 min. Then the probability of waiting less than 45 min is: P (waiting more than 45 min) is: Event Sample Space Session 10 Operations Management

Probability Distribution for Discrete Random Variables Let us begin with discrete outcomes A probability distribution is a list of: All possible values for a random variable (Sample space); and The corresponding probabilities For a die, the probability distribution is: Outcome Probability 1 1/6 2 3 4 5 6 Session 10 Operations Management

Probability Distribution for Discrete Random Variables The chart below depicts the probability distribution Session 10 Operations Management

Cumulative Probability Distribution for Discrete Random Variables Probability that a random number will be less than or equal to some given number For a die, the cumulative probability distribution is: Outcome less than or equal to: Probability 1 1/6 2 2/6 3 3/6 4 4/6 5 5/6 6 Additional: What is the probability a die roll is less than 3.5? Session 10 Operations Management

Continuous Random Variables and Probability Density Functions (PDF) The probability density function is the analog of the probability distribution (table 1) for discrete random numbers Example: Suppose we have a computer program that can generate any number between 1 and 6 (not just the integers) Assume that each number is equally likely to be generated. Then we have a continuous random number This random number has a uniform distribution between 1 and 6 Session 10 Operations Management

Continuous Random Variables and Probability Density Functions (PDF) 1/5 1 2 3 4 5 6 Outcome Session 10 Operations Management

Properties of Probability Density Functions By convention the total area under the probability density function must equal 1 The base of the rectangle in the figure is 6 – 1 = 5 units long, the probability density is 1/5 for all values between 1 and 6. This ensures that the total area is 1 The probability of observing any value between two numbers is equal to the area under the probability density function between those numbers The probability of observing any number between 4.0 and 5.0 will be (5.0 – 4.0)* 1/5 = 1/5 = 0.2 Session 10 Operations Management

Properties of Probability Density Functions 1/5 1 2 3 4 5 6 Outcome Session 10 Operations Management

Properties of Cumulative Distribution Functions CDF, F Probability that the outcome is smaller than 5: is 4/5 1 4/5 Probability that the outcome is smaller than 2: is 1/5 1/5 1 2 3 4 5 6 Outcome Session 10 Operations Management

Relationship between Density and Cumulative Distribution Functions CDF 1 4/5 1/5 1 2 3 4 5 6 Outcome Session 10 Operations Management

Other distributions: Triangular Probability that the outcome is between 2 and 5 2 5 Session 10 Operations Management

Operations Management Normal Distribution 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 -4 -3 -2 -1 1 2 3 4 X Normal distribution #1 Normal distribution #2 Session 10 Operations Management

Continuous Random Variables and Probability Density Functions (PDF) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 -4 -3 -2 -1 1 2 3 4 X Normal distribution #1 Normal distribution #2 Session 10 Operations Management

Cumulative Density Function (CDF) for Continuous Random Numbers This is analogous to the cumulative distribution function for discrete random numbers The cumulative density function gives the probability of the continuous random variable being equal to or smaller than a given number Session 10 Operations Management

Cumulative Density Function (CDF) for Continuous Random Numbers 1 1/5 o 1 2 3 4 5 6 Session 10 Operations Management

Mean or Expected Value of a Random Number Expected value can be thought of as the average value of a random variable Let us denote by X the value of the random variable. If the random variable is the value of a die, then X denotes the value rolled. If we roll a 6, then X = 6). We will use the notation E[X] to denote the expected value of X If the random number is a discrete variable that can take on values between 1 and N then: E[X] = Thus for the die, E[X] = 1/6*1 + 1/6*2 + 1/6*3 + 1/6*4 + 1/6*5 + 1/6*6 = 3.5 Session 10 Operations Management

Mean or Expected Value of a Random Number What if the variable is a continuous random variable? Let f(X) be the probability density function. Example: for the uniform distribution, we have seen: f(X) = 0.2 whenever X is between 1 and 6. f(X) = 0 if X is not between 1 and 6.] Integration of continuous variables in lay terms is equivalent to summation for discrete variables. Session 10 Operations Management

Operations Management The Variance of X When X is a discrete random variable: Var(X) =  (X – E[X])2*Prob(X) If X is the random number generated by the roll of a die then: Var(X) = (1-3.5)2*1/6 + (2-3.5)2*1/6 +(3-3.5)2*1/6 +(4-3.5)2*1/6 +(5-3.5)2*1/6 +(6-3.5)2*1/6 = 2.9166 Standard Deviation = square root of variance SD(X) = 1.708 in this example Session 10 Operations Management

How to measure variability? A possible measure is variance, or standard deviation Is this good enough? Session 10 Operations Management

Which one has the larger variability? Session 10 Operations Management

Which one has the larger variability? The variation in the first set appears to be significantly higher than the second set. Nevertheless, the standard deviation of the first graph is 5, the standard deviation of the second graph is 10. Session 10 Operations Management

Coefficient of Variation A better measure of variability is the ratio of the standard deviation to the average. This ratio is called the coefficient of variation. Coefficient of Variation = Standard Deviation / Average (expected value) A similar measure is squared coefficient of variation: SCV = (CV)2 = (SD/M)2 Session 10 Operations Management

Operations Management Sum of Random Numbers Often we have to analyze sum of random numbers. Examples include: The sum of the demand of different products processed by the same resource The total demand for cars produced by GM The total demand for knitwear at DD The total completion time of a project The sum of throughput times at two different stages of a service system (waiting time to place an order at a cafeteria and waiting time in the line to pay for the food) Session 10 Operations Management

Operations Management Sum of Random Numbers Let X and Y be two random variables. The sum of X and Y is another random variable. Let S = X +Y The distribution of S will be different from that of X and Y Example: Let S be the sum of the values when you roll 2 dice simultaneously. Let X represent the value die #1 and Y represent the value of die #2 S = X + Y Session 10 Operations Management

Operations Management Sum of Random Numbers The distribution of the sum S is given below: S Prob(S) 2 1/36 7 6/36 3 2/36 8 5/36 4 3/36 9 4/36 5 10 6 11 12 Session 10 Operations Management

Operations Management Sum of Random Numbers E[S] = 2*1/36 + 3*2/36 + 4*3/36 + 5*4/36 + 6*5/36 + 7*6/36 + 8*5/36 + 9*4/36 + 10*3/36 + 11*2/36 + 12*1/36 = 7 Var(S) = (2 - 7)2*1/36 + (3 - 7)2*2/36 +…….+ (12 - 7)2*1/36 = 5.83 SD(S) = 5.83^1/2 = 2.42 Session 10 Operations Management

Operations Management Sum of Random Numbers Session 10 Operations Management

Expected Value and Standard Deviation of Sum of Random Numbers If a and b are 2 known constant and X and Y are random independent variables: E[aX+bY] = aE[X] + bE[Y] Var(aX+bY) = a2Var(X) + b2Var(Y) Session 10 Operations Management

Specific Distributions Of Interest We will also utilize Uniform Distributions Uniform Distribution: Whenever the likelihood of observing a set of numbers is equally likely Continuous or discrete We use notation U(a,b) to denote a uniform distribution Example U(1,5) is uniform distribution between 1 and 5. If it is a discrete distribution then outcomes 1,2,3,4, and 5 are equally likely (each with probability 1/5) If it is a continuous distribution then all numbers between 1 and 5 are equally likely The p.d.f. for U(1,5) (continuous) will be f(X) = 0.25 for X between 1 and 5 Session 10 Operations Management

Exponential Distribution The exponential distribution is often used as a model for the distribution of time until the next arrival. The probability density function for an Exponential distribution is: f(x) = e-x, x > 0  is a parameter of the model (just as m and s are parameters of a Normal distribution) E[X] = 1/ Var(X) = 1/2 Coefficient of Variation = Standard deviation / Average = 1 Session 10 Operations Management

Exponential Distribution Shape of the Exponential Probability Density Function f(X) X Session 10 Operations Management

Operations Management Poisson Distribution The Poisson Distribution is often used as a model for the number of events (such as the number of telephone calls at a business or the number of accidents at an intersection) in a specific time period The probability of n events is: p(n) = ne-/n!, n = 0, 1, 2, 3, …  is a parameter of the model E[N] =  Var(N) =  Session 10 Operations Management

Operations Management Poisson Distribution Session 10 Operations Management

Operations Management Next Class Waiting-line Management How uncertainty/variability and utilization rate determines the system performance Article Reading: “The Psychology of Waiting-lines” Session 10 Operations Management