A1 45730 24sep08

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 fx x λ =≥ Cumulative distribution function: ( ) 1 for 0 x Fx e x =− Expected value: 1 () EX = Variance: 2 1 var( ) X = [= [= 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 = • 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 11 e x p 2 2 x fx πσ =− Mean: E [ X ] = Variance: var( X ) = 2 Shorthand: X ~ N ( , 2 ) try to integrate this sucker!!
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Probability distribution function for a particular normal random variable: X ~ N (4,1) (Mean = 4, Variance = 1) -2 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
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Probability distribution function for another normal random variable: X ~ N (4,3) (Mean = 4, Variance = 3) -2 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
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This note was uploaded on 10/05/2010 for the course BUS 45730 taught by Professor Ravi during the Fall '08 term at Carnegie Mellon.

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

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