Chapter 2.8-2.9 - Math3200 Intermediate Probability and Statistics Prof Nan Lin Department of Mathematics Washington University CONTINUOUS DISTRIBUTIONS

Chapter 2.8-2.9 - Math3200 Intermediate Probability and...

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Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University
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CONTINUOUS DISTRIBUTIONS Uniform Exponential Gamma Beta Normal
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Uniform Distribution U[a,b] It is described by the p.d.f.: f(x) x b a area = width x height = (b a) x = 1
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8.4 Example The amount of gasoline sold daily at a service station is uniformly distributed with a minimum of 2,000 gallons and a maximum of 5,000 gallons. What is the probability that the service station will sell at least 4,000 gallons? Algebraically: what is P(X ≥ 4,000) ? P(X ≥ 4,000) = (5,000 4,000) x (1/3000) = .3333 f(x) x 5,000 2,000
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Uniform distribution Mean: E ? = ? + ? 2 Variance 𝑉?? ? = ? − ? 2 12 p.d.f. c.d.f.
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Exponential distribution ?𝑥?(?) Waiting for an elevator. Most likely, the experience may be it frequently comes in a short while and once in a while, it may come pretty late. If we want to use a random variable to measure the waiting time for elevator to come, 1. It must be continuous. 2. Smaller values have larger probability and larger values have smaller probability. 0 0 0.2 0.4 0.6 0.8 1 x R(x)
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Poisson Exponential If the number of occurrences in a given time interval follows a Poisson distribution with parameter ? , then the waiting time between occurrences is exponentially distributed with parameter ? . ? : the average number of occurrences per unit time If ? denotes the number of events which happen in unit time, then More generally, the probability that exactly 𝑘 events will occur in time interval 𝑇 is Let ? denote the time between successive events, then ? > 𝑇 means that the first arrival takes more than T units of time, i.e. there are 0 arrivals in the first 𝑇 units of time. So, 𝑃 ? > 𝑇 = 𝑃 ? = 0 = ?T 0 0! ? −𝜆𝑇 = ? −𝜆𝑇
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Exponential distribution c.d.f. ? 𝑥 = 1 − ? −𝜆𝑥 , 𝑥 ≥ 0 p.d.f. ? 𝑥 = ?? −𝜆𝑥 , 𝑥 ≥ 0 Mean: E ? = 1 ? Variance 𝑉?? ? = 1 ? 2 p.d.f. c.d.f.
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Arrivals of babies at a lying-in hospital Arrivals of babies are independent events. (The mothers don’t conspire together.) The probability of babies being born almost simultaneously is negligible. The probability of a baby being born in a time interval only depends on the length of the time interval, not on when it is during the day. Fact : For many years the hospital has been averaging 6 births per day. If ? denotes the number of babies born in a day ? ? = 6 babies/day If ? denotes the time between births ? ? = 1 6 days/baby
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Arrivals of babies at a lying-in hospital Given that a baby has just been born, what is the probability that the next baby will take one or more hours to arrive? Convert hours to days
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Arrivals of babies at a lying-in hospital After a baby was born, the next baby did not arrive after 3 hours. What is the probability that the next baby will take one or more hours to arrive?
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