M670_4(Discrete Distributions)

# M670_4(Discrete Distributions) - Discrete Probability...

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1 Discrete Probability Distributions k MGMT 670: Business Analytics Krannert School of Management Purdue University

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2 Random Variables § Numerical description of the outcome of an experiment § Classified as either  discrete  or  continuous § Discrete random variable : either a finite number of  values or an infinite sequence of values such as 0, 1, 2,  3, …. § Continuous random variable : any numerical value in  an interval or collection of intervals.
3 Examples of Discrete Random Variables § Experiment : Make 100 sales calls, and record the  number of sales made. Possible values: 0, 1, 2, 3, …, 100 § Experiment : Operate a restaurant for one day, and  record the number of customers entered. Possible values: 0, 1, 2, 3, … § Experiment : Sell an automobile, and record the  gender of the customer. Possible values: male, female

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4 Examples of Continuous Random Variables § Experiment : Operate a bank, and record the time  between customer arrivals ( X ). Possible values:   X  ‡  0. § Experiment : Observe a machine’s working hours, and  record the utilization rate in an eight-hour workday  ( Y ). Possible values:  0 £   Y  £  100%
5 Discrete Probability Distributions § List of all possible pairs of ( xi, f(xi  ))  where xi  = a value of the random variable  X f ( xi ) = probability of getting value of  xi § f ( xi ) is referred to as the  probability mass function . § Conditions for the probability mass function: § Described with a table, graph, or equation . 1 ) ( 1 ) ( 0 = i i i x f x f

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6 Example: Oil Commodities § A commodities investor is concerned with the price of oil  for the coming year. § The investor believes there are four possible scenarios for  the oil market in the coming year: high demand, moderate  demand, no growth, or moderate contraction. § She estimates that the price of oil per barrel in each case  will be \$78, 73, 63, and 50, respectively. § Also, she has assessed that the probabilities of these  outcomes are 0.10, 0.50, 0.25, and 0.15,                respectively.
7 Distribution of Oil Price (X) Economic Outcome Price,  xi Probability High Demand 78 0.10 Moderate Demand 73 0.50 No Growth 63 0.25 Moderate Contraction 50 0.15 Total 1.00

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8 Expected Value of Discrete Random Variable § Weighted average of all possible values = = i i i x f x X E ) ( ) ( μ x i p ( x i ) x i  p ( x i ) 78 0.10  7.80  73 0.50  36.50  63 0.25  15.75  50 0.15  7.50  Sum 1.00  67.55  E ( X )
9 Variance of Discrete Random Variable § Weighted average of squared deviations from the mean ( 29 ( 29 2 2 2 Var ( ) ( ) i i i (X)σ E X E X x E X f(x ) = = - = - x i p ( x i ) x i   p ( x i ) ( x i  -  μ ) 2 ( x i  -  μ ) 2 p ( x i ) 78 0.10  7.80  109.2025 10.920  73 0.50  36.50  29.7025 14.851  63 0.25  15.75  20.7025 5.176  50 0.15  7.50  308.0025 46.200  Sum 1.00  67.55  77.148  σ 2

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10 Covariance § Measure of the  linear association
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