Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 1
Triangular Probability Distributions involve generating random variables based on specified parameters. By understanding the concepts of height, area, and likelihood, one can effectively compute values and make informed decisions in various scenarios. This content delves into the calculation methods and practical applications of triangular distributions.
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Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 1
Basic Probability Distributions How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality Albert Einstein Some parts of these slides were prepared based on Essentials of Modern Busines Statistics, Anderson et al. 2012, Cengage. Managing Business Process Flow, Anupindi et al. 2012, Pearson. Project Management in Practice, Meredith et al. 2014, Wiley
Triangular Probability Distribution
Triangular Probability Distributions a m b a = smallest value the variable can assume b = largest value the variable can assume m = most likely value the variable can assume Expected Value of x = (a +m+ b)/3 Variance of x s 2 = (a2 + m2 +b2 ab- am-bm)/18 Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 4
Generate a Triangular Random Variable Compute the height of this triangle 1 m a = Area = ( ) m a L ( ) b a b a H/2= 1/(b-a) a m b Similarly, the area to the right is 1 b m = Area = ( ) b m R ( ) b a b a Generate a random number if rand() (m-a)/(b-a) we are on the left, otherwise on the right Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 5
Triangular Probability Distributions Suppose we are on the left Then the triangular random variable is something like X H h a m b X What is the height associated with X? X a h X a X a = = = h h H ( )( ) m a b a H m a m a = ( ) AreaLX h X a The area to the left of X is then Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 6
Triangular Probability Distributions = ( ) AreaLX h X a X a H = h ( )( ) m a b a h a m b X 2 ( ) X a = AreaLX ( )( ) m a b a The area to the left of X was our rand() which happened to drop on the left 2 ( ) X a = () Rand ( )( ) m a b a Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 7
Triangular Probability Distributions 2 ( ) X a = () Rand ( )( ) m a b a H h a () m b X = 2 ( ) ( )( ) X a m a b a Rand = + ( )( ) () X a m a b a Rand Generate a random number if (m-a)/(b-a) we are on the left, and implement the above equation to generate the random variable, otherwise we are on the right Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 8
Triangular Probability Distributions We showed that for X on the left H 2 ( ) X a = AreaLX h ( )( ) m a b a a m b X Similarly, we can show that if X is on the right of m, the area to the right of h is 2 ( ) b X = AreaRX ( )( ) b m b a This is the area to the right of X, but the area to the left of X was equal to Rand() The area to the right of X is equal to 1- Rand() Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 9
Triangular Probability Distributions 2 ( ) b X = AreaRX ( )( ) b m b a H h 2 ( ) b X = 1 () Rand a m b X ( )( ) b m b a = 2 ( ) ( )( )( 1 ()) b X b m b a Rand = ( )( )( 1 ()) X b b m b a Rand Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 10
Triangular Probability Distributions Generate a random number if (m-a)/(b-a) we are on the left, otherwise on the right. = + ( )( ) () X a m a b a Rand If on the left If on the Right = ( )( )( 1 ()) X b b m b a Rand Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 11
Triangular Probability Distributions 10 5 m a Suppose a=5, b= 20, m=10 = = = 333333 . 0 AreaL 20 5 b a Generate a random number If Rand() 0.333333 = + 5 75 () X Rand = + ( )( ) () X a m a b a Rand Otherwise = = ( )( )( 1 ()) 20 150 1 ( ()) X b b m b a Rand X Rand Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 12
Generate Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 13
Histogram Histogram 160 140 120 100 Frequency 80 Frequency 60 40 20 0 5 6 7 8 9 10 11 12 13 Bin 14 15 16 17 18 19 20 More Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 14
Descriptive Statistics Mean Standard Error Median Mode Standard Deviation 3.09 Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count 11.62 0.10 11.32 #N/A 9.53 -0.57 0.31 14.30 5.43 19.73 11624 1000 Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 15
Practice: Simulate in excel: All Activities Follow Triangular Distribution 6,4,7 3,4,6 A3 E A1 3,4,5 2,3,5 S A4 A6 1,3,5 1,2,4 11 8 10 A2 A5 What is the probability of completing the project in 12 days? Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 16
Random Variable Generation Distributi on Uniform Normal Exponential Mean Var Generation (a+b)/2 (b-a)/12 x=a+(b-a)Rand() x= + NORMSINV(Rand()) x=- LN(Rand()) If Rand() (m-a)/(b-a) Triangle (a2 + m2 +b2 ab- am-bm)/18 (a +m+ b)/3 = + ( )( ) () x a m a b a Rand Otherwise b x = ( )( )( 1 ()) b m b a Rand Basics Probability Distributions- Uniform Ardavan Asef-Vaziri Jan.-2016 17
