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ECE 307 – Techniques for Engineering Decisions Probability Distributions George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved OUTLINE OF DISTRIBUTION REVIEWED ‰ Discrete  Binomial  Poisson ‰ Continuous  Exponential  Normal © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved THE BINOMIAL DISTRIBUTION ‰ Binomial distributions are used to describe events with only two possible outcomes ‰ Basic requirements are  dichotomous outcomes: uncertain events occur in a sequence with each event having one of two possible outcomes such as © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved THE BINOMIAL DISTRIBUTION success/failure, correct/incorrect, on/off or true/false  constant probability : each event has the same probability of success  independence: the outcome of each event is independent of the outcomes of any other event © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved BINOMIAL DISTRIBUTION EXAMPLE ‰ We consider a group of n identical machines with each machine having one of two states: P {machine is on} = p P {machine is off } = q = − p ‰ For concreteness, we set n = and define for i = 1, 2, … , 8, the r.v s © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved BINOMIAL DISTRIBUTION EXAMPLE ⎧⎪ Xi =⎨ ⎪⎩ machine i is on with prob p machine i is off with prob q = − p ‰ The probability that or more machines are on is determined by evaluating ⎧n ⎫ P ⎨∑ X i ≥ ⎬ = P {3 or more machines are on} ⎩ i =1 ⎭ © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved BINOMIAL DISTRIBUTION EXAMPLE = P {3 machines are on} + P {4 machines are on} + + P {8 machines are on} ⎧n ⎫ P ⎨∑ X i ≥ ⎬ = ⎩ i =1 ⎭ 8! ∑ ( − r ) !r! p r (1 − p ) 8− r r=3 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved THE BINOMIAL DISTRIBUTION ‰ In general, for a r.v R with dichotomous outcomes of success and failure, the probability of r successes in n trials is P { R = r in n trials with probability of success p} n! n−r r = p (1 − p ) (n − r ) !r ! the binomial distribution © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved THE BINOMIAL DISTRIBUTION ‰ We can show that: E { R} = np var { R} = np ( − p ) ⎧n ⎫ P ⎨∑ X i ≥ k ⎬ = ⎩ i =1 ⎭ n n! ∑ ( n − r ) !r! p (1 − p ) r n-r r =k © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved EXAMPLE: SOFT PRETZELS ‰ Pretzel entrepreneur can sell pretzels at $ 0.50 per unit with a market potential of 100,000 pretzels within a year; there exists a competing product and so we know he cannot sell that many ‰ Basic model is binomial: new pretzel is a hit ⇒ (success) new pretzel is a flop (flop) captures 30% of market in one year ⇒ captures 10% of market in one year © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 10 CASE STUDY: OVERBOOKING ‰ We can show that for reservations = 19 E {π 19 res} < 1180.59 ‰ We conclude that the profits are maximized if reservations = 17 and so any greater overbooking is at the sacrifice of lower profits © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 65 CASE: MUNI SOLID WASTE ‰ This case concerns the “risks” posed by constructing an incinerator for disposal of a city’s solid waste ‰ There is no question of “whether” to construct an incinerator since landfill was to be full within years and no other choices are apparent © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 66 CASE: MUNI SOLID WASTE ‰ Of particular interest are the residual emissions that need to be estimated for  dioxins  furans (organic compounds)  particulate matter  SO2 (represents acid gases) © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 67 CASE: MUNI SOLID WASTE ‰ The specifications for a 250 ton/day incinerator – borderline between the EPA – classified small and medium plants – have to meet the EPA’s proposed emission levels for the three key pollutants  dioxins/furans – denoted by D  particulate matter – denoted by PM ~  sulphur dioxide – denoted by SO ~ © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 68 CASE: MUNI SOLID WASTE plant capacity in tons of waste per day pollutant dioxins/furans mg/nm PM (mg/dscm) ~ SO (ppmdv) ~ small (below 250) medium (above 25) 500 125 69 69 – 30 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 69 CASE: MUNI SOLID WASTE ‰ The typical approach in environmental risk anal- ysis is a “worst case” scenario assessment which fails to capture the uncertainty present in both the amount of waste and the contents of the waste © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 70 CASE: MUNI SOLID WASTE ‰ The lognormal distribution is used to represent the distribution for emission levels ‰ Lognormal distribution parameters μ and σ for pollutants pollutant r.v μ σ D 3.13 1.20 PM 3.43 0.44 ~ SO2 3.20 0.39 ~ © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 71 CASE: MUNI SOLID WASTE lognormal density function for SO2 emissions from incineration plant probability 10 20 30 40 50 60 70 80 SO2 level (ppmdv) © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 72 CASE: MUNI SOLID WASTE ‰ We define the r.v X , which is lognormally distributed with parameters μ and σ , by Y = ln ( X ) ~ N ( μ , σ ) E {X} = e ⎛ 2⎞ ⎜μ + σ ⎟ ⎝ ⎠ and var { X } = e 2μ (e σ ) −1 e σ2 ‰ We evaluate the probability of exceeding the small plant levels of emissions © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 73 CASE: MUNI SOLID WASTE P { D > 500} = P {Y = ln D > ln ( 500 )} Y − 3.13 Z = 1.2 = ⎧ P ⎪⎨ Z ⎩⎪ affine transformation ln ( 500 ) − 3.13 ⎫⎪ > ⎬ 1.2 ⎪ = P { Z > 2.57} ⎭ 0.0051 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 74 CASE: MUNI SOLID WASTE { } { } P PM ~ > 69 = P Y = ln ( PM ~ ) > ln ( 69 ) Y − 3.43 Z = 0.44 = ⎧ P ⎪⎨ Z ⎩⎪ affine transformation ln ( 69) − 3.43 ⎫⎪ > ⎬ 0.44 ⎪ ⎭ 0.0336 = P { Z > 1.83} © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 75 CASE: MUNI SOLID WASTE P ⎧⎨ SO ⎩ ~ > 30 ⎫⎬ ⎭ = ⎧⎪ P ⎨Y ⎪⎩ Y − 3.20 Z = 0.39 = ⎧ P ⎪⎨ Z ⎪ ⎩ { = ⎫ ln ⎨⎧ SO ⎬ ⎩ ~ 2⎭ > ln ( 30 ) ⎫⎪ ⎬ ⎭⎪ affine transformation ln ( 30) − 3.20 ⎫⎪ > ⎬ 0.39 ⎪ ⎭ } = P Z > 0.52 0.3015 the probability of a single observation © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 76 CASE: MUNI SOLID WASTE ‰ In practice, SO has to be monitored continuously ~ and the average daily emission level must remain below the level specified in the table ‰ If we take 24 hourly observations X of SO ~ ~i levels and define the geometric mean ⎡ n ⎤ G = ⎢∏ X i ⎥ ⎣ i =1 ⎦ n , n = 24 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 77 CASE: MUNI SOLID WASTE then, we can show that G is also lognormal with σ μ , parameters ⎛ ⎜ ⎜ ⎝ P ⎧ ⎨G ⎩ > ⎫⎬ ⎭ Z = 24 = P ⎞ ⎟ ⎟ ⎠ ⎧⎪ ⎨Y ⎩⎪ = ln Y − 3.20 > ln ⎛ ⎞ ⎫⎪ ⎜ 30 ⎟⎬ ⎝ ⎠ ⎭⎪ affine transformation 0.39 / 24 = P ⎛ ⎞ G ⎜ ⎟ ⎝ ⎠ ⎧ ⎪ Z − ⎨ ⎪⎩ 24 ⎫ ⎪ > ⎬ ⎭⎪ 0 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 78 CASE: MUNI SOLID WASTE ‰ Note that this is a much smaller probability than for a single observation and leads to a more realistic assessment of the probability ‰ The requirement of a small plant are therefore met © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 79 ... Urbana-Champaign, All Rights Reserved THE BINOMIAL DISTRIBUTION ‰ We can show that: E { R} = np var { R} = np ( − p ) ⎧n ⎫ P ⎨∑ X i ≥ k ⎬ = ⎩ i =1 ⎭ n n! ∑ ( n − r ) !r! p (1 − p ) r n-r r =k... Illinois at Urbana-Champaign, All Rights Reserved EXAMPLE: SOFT PRETZELS ‰ Pretzel entrepreneur can sell pretzels at $ 0.50 per unit with a market potential of 100,000 pretzels within a year; there... University of Illinois at Urbana-Champaign, All Rights Reserved 16 REQUIREMENTS FOR A POISSON DISTRIBUTION ‰ Events can happen at any of a large number of values within the range of measurement