b a s a m b Continuous Probability Distributions Exponential Distribution Uniform Distribution • Triangular Distribution • Normal Distribution
Continuous Probability Distributions • The probability of the continuous random variable assuming a specific value is 0. • The probability of the random variable assuming a value within a given interval from x1 to x2 is defined to be the area under the graphof the probability density functionbetween x1and x2.
Uniform Probability Density Distribution Uniform Probability Density Function f (x) = 1/(b - a) for a<x<b = 0 elsewhere a = smallest value the variable can assume b = largest value the variable can assume b a µ = (a + b)/2 s 2 = (b - a)2/12
b b a a x1 x2 x1 Continuous Probability Distributions b a x1 P(x1≤x≤ x2) P(x≤ x1) P(x≥ x1)= 1- P(x<x1) P(x =x1)= 0 P(x≥ x1)
Example Slater customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces. Probability Density Function f (x ) = 1/10 for 5≤ x ≤ 15 = 0 elsewhere where x = salad plate filling weight
F (x ) P(12 <x< 15) = 1/10(3) = .3 1/10 x 5 10 12 15 Salad Weight (oz.) Example What is the probability that a customer will take between 12 and 15 ounces of salad?
The Uniform Probability Distribution P(8 ≤ x ≤ 12) = ? f (x ) 1/10 x 5 8 12 15 P(8 ≤ x ≤ 12) = (1/10)(12-8) = .4
The Uniform Probability Distribution P(0<x< 12) = ? f (x ) 1/10 x 5 12 15 P(0<x< 12) = P(5<x < 12)= = (1/10)(12-5) = .7
Generate a Uniform Random Number • We want to select a student, randomly, to come to the white board. • How? • Simulation is based on random number generation • Crystal Ball is a simulation software • AutoMod is another simulation software Ardavan Asef-Vaziri
Show Some AutoMod Examples 11/16/2014 Ardavan Asef-Vaziri 1-10
Triangular Probability Distributions a x1 m b a = smallest value the variable can assume b = largest value the variable can assume m = most likely value the variable can assume Compute the height of this triangle If a ≤ x1 ≤ m, compute the height associated with x1. You can compute it. But you need time.
Triangular Probability Distributions a m b x1 It is equal to 2(x1-a)/[(m-a)(b-a)]. Compute P (x ≤ x1). It is equal to the area from a to x1 It is equal to (x1-a)2/[(m-a)(b-a)]. It is not difficult to compute the area if m ≤ x1 ≤ b,
Triangular Probability Distribution Expected Value of x µ = (a +m+ b)/3 Variance of x s 2 = (a2 + m2 +b2 – ab- am-bm)/18
Exponential Probability Distribution m = mean s = m e = 2.71828182845904, the base of the natural logarithm.
F (x ) .4 P(x< x1 ) = area from 0 to x1 .3 .2 .1 x 1 2 3 4 5 6 7 8 9 10 Exponential Probability Distribution
F (x ) .4 .3 .2 .1 x 1 2 3 4 5 6 7 8 9 10 Example The time between arrivals of cars at Al’s Carwash follows an exponential probability distribution with a mean time between arrivals of 3 minutes. Al would like to know the probability that the time between two successive arrivals will be 2 minutes or less. P(x< 2) = 1 - e-2/3 = 1 - .5134 = .4866
Generate an Exponential l Random Variable • Parameters: Mean and StDev. • How can I generate an exponential random variable? Ardavan Asef-Vaziri
Normal Distribution z μ Ardavan Asef-Vaziri
Normal Probability Density Function where = mean = standard deviation = 3.14159 e = 2.71828
Standard Normal Probability Density Function where = 0 = 1 = 3.14159265358979
Normal and Standard Normal A Normal Variable x is associated with a standard Normal Variable z x is Normal (Average x , Standard Deviation x) z is Normal (0,1) μ = Average x σ = Standard Deviation of x is z a Standard Normal variable. Ardavan Asef-Vaziri
Standard Normal Table s m x x-scale z-scale 0 z • There is a table for z which tells us • Given anyprobability of not exceeding z, what the value of z is • Given anyvalue forz, what the probability of not exceeding z is Ardavan Asef-Vaziri
Table returns probability Given z Probability Given z, Find the Probability 0.05 z Second digit after decimal Up to the first digit after decimal 1.6 z = 1.65
Given Probability, Find z Given a 95% Probability The table will give you z Probability 0.05 z Second digit after decimal Up to the first digit after decimal 1.6 Z = 1.65
F(z) z 0 The Standard Normal Distribution F(z) F(z) = Prob( N(0,1) <z)
Example Pep Zone sells auto parts and supplies including multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. The store manager is concerned that sales are being lost due to stockouts while waiting for an order. It has been determined that lead time demand is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. In Summary; we have a N (15, 6): A normal random variable with mean of 15 and std of 6. The manager would like to know the probability of a stockout, P(x > 20).
Example If the manager of Pep Zone wants the probability of a stockout to be no more than .05, what should the reorder point be? Area = .05 Area = .95 z.05 0
Generate Normal Random Varaiable • Parameters: Mean and StDev. • How can I generate a standard Normal random variable? • How can I generate a Normal random variable? • Central Limit Theorem Ardavan Asef-Vaziri
A Random Variable = Summation of Several Random Variables We have a random variable y which is equal to summation of n random variables x: N(mx, σx) y = x1+x2+x3+…….+xn Find average and standard deviation of the random variable y in terms of mx, σx Note: on the basis of CLT, if n is large, we do not need N, and other distribution also leads to Normal distribution for y Ardavan Asef-Vaziri
A Random Variable = Summation of Several Random Variables Mean (y) = Mean(x1)+ Mean(x2)+ ….. +Mean(xn) my = mx1+…….mxn Var(y )= Var(x1)+Var(x2)+ …+Var(xn) sy2 = sx12 +…….s2 xn StdDev (y) = SQRT(Var(y)) Ardavan Asef-Vaziri
A Random Variable = Another Random Variable Multiplied by a Constant We have a random variable y which is equal to random variables x: N(mx, σx) multiplied by a constant k y = kx Find Average and Standard deviation of the random variable y in terms of mx, σx Ardavan Asef-Vaziri
A Random Variable = Another Random Variable Multiplied by a Constant Mean (y) = kMean(x) my = kmx Var(y )= k2Var(x) sy2 = k2sx2 StdDev (y) = SQRT(Var(y)) Ardavan Asef-Vaziri
Additional Assignment for next week Problem 1. Initial investment at the end of year 0 is 100, cash inflow at the end of year 1 and 2 each N(100,20). Compute mean and standard deviation of the NPV. Compute Probability of NPV ≥ 0. 2. A project has three sequential tasks. The first task follows Exponential distribution with parameter of 5 days. The second task follows Uniform distribution with parameters of 5 and 10 days, the last task follows Normal distribution with parameters of 9 and 2 days. The numbers on the next page were generated using RAND() function in excel. Starting from the first row and column, moving through the first row first, and not using any given element twice, generate a random instance of duration of this project. 11/16/2014 Ardavan Asef-Vaziri 1-35
Example By not using any element twice we mean not using a specific element – in a specific row and column twice. For example you can not use the number in row 1 and column 1 twice. Obviously you can use all the three 0.1s as soon as you reach their corresponding box in the table. 11/16/2014 Ardavan Asef-Vaziri 1-36