A1 45730 24sep08 - 45 730 Probability Decision Making...

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45 730 Probability & Decision Making (9/24/08) Topics for today’s class: Continuous Probability Distributions • Exponential • Normal
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The exponential distribution: Situation: Examining the length of “time” until an event occurs Examples: Time until next arrival at service station Time until next call to a 1-800 number Length until next flaw in sheet metal
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The probability model for an exponential r.v. Sample space: [0, ) Additional parameter: λ “arrival rate” Probability distribution function: ( ) e for 0 x f x x λ λ = Cumulative distribution function: ( ) 1 for 0 x F x e x λ = Expected value: 1 ( ) E X λ = Variance: 2 1 var( ) X λ = [= [= EXPONDIST(x EXPONDIST(x , λ ,0)] ,0)] [= [= EXPONDIST(x EXPONDIST(x , λ ,1)] ,1)]
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The probability distribution function for an exponential random variable: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 10
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A close relationship: Poisson and exponential random variables Exponential: Time to next arrival Average time between arrivals = 1/ λ Units: time Poisson: Number of arrivals Arrival rate = λ Units: 1/time Exponential distribution flips Poisson distribution to examine time of arrival rather than number of arrivals
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Example: Calls to 911 Suppose 911 system averages 10 calls per hour Implications: Call arrivals per hour are Poisson r.v. with λ = 10 Time until next call (in hours) is exponential r.v. Questions: What is expected time (in minutes) until next call? What is probability that next call will arrive in next 5 minutes?
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Expected time to next call: Probability that next call arrives within 5 minutes:
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Suppose no call in first 5 mts.What is the probability that next call is after 10 mts? What is the probability that next call is after five minutes?
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The normal distribution: Situation: Outcomes can fall between - and Outcomes are “most likely” around the mean and less likely in the “tails” (i.e. extreme values) Probabilities for outcomes are symmetric around mean symmetric bell-shaped probability model Examples: test scores asset prices weights of people, products, etc. various other applications in nature and business
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Why is the normal distribution so important? Many random variables exhibit outcome patterns that can be described by normal probability model Normal random variables have many convenient properties As we will see later (next class and especially in the next mini), many important results in statistics involve the normal distribution (Central Limit Theorem)
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The probability model for normal random variables The normal distribution is completely described by two parameters ( μ , σ 2 ) Probability distribution fct: ( ) 2 2 2 1 1 ( ) exp 2 2 x f x μ σ πσ = Mean: E [ X ] = μ Variance: var( X ) = σ 2 Shorthand: X ~ N ( μ , σ 2 ) try to integrate this sucker!!
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