Lecture10 - lecture 10, Discrete Random Variables III...

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Unformatted text preview: lecture 10, Discrete Random Variables III & Continuous Random Variable I 1/26 Poisson Distribution Continuous Probability Distributions The Uniform Distribution lecture 10, Discrete Random Variables III & Continuous Random Variable I Outline 1 Poisson Distribution 2 Continuous Probability Distributions 3 The Uniform Distribution lecture 10, Discrete Random Variables III & Continuous Random Variable I 2/26 Poisson Distribution Continuous Probability Distributions The Uniform Distribution lecture 10, Discrete Random Variables III & Continuous Random Variable I Poisson Distribution Motivating Examples • What’s the distribution of the number of calls arriving at the help desk in a 30-minute interval? • What’s the distribution of the number of times a web server is accessed per hour? • What’s the distribution of the number of traffic accidents in a city per day? • What’s the distribution of the number of imperfections per square meter of glass panel • ... lecture 10, Discrete Random Variables III & Continuous Random Variable I 3/26 Poisson Distribution Continuous Probability Distributions The Uniform Distribution lecture 10, Discrete Random Variables III & Continuous Random Variable I Poisson Distribution The Poisson Distribution • Consider the number of times an event occurs over an interval of time or a region of space, and assume that 1. The probability of occurrence is the same for any intervals of equal length 2. Whether the event occurs in any interval is independent of whether the event occurs in any other non-overlapping interval • If X is the number of occurrences in a specified interval, then X is a Poisson random variable . lecture 10, Discrete Random Variables III & Continuous Random Variable I 4/26 Poisson Distribution Continuous Probability Distributions The Uniform Distribution lecture 10, Discrete Random Variables III & Continuous Random Variable I Poisson Distribution The Poisson Distribution, ctd • Suppose λ is the mean or expected number of occurrences during a specified interval, then the probability of x occurrences in the interval is given by the Poisson distribution p ( x ) = e- λ λ x x ! , x = , 1 ,..., where e ≈ 2 . 718282 is the base of the natural logarithms. • The number of occurrences X is said to be a Poisson random variable with parameter λ , or simply a Poisson ( λ ) random variable. lecture 10, Discrete Random Variables III & Continuous Random Variable I 5/26 Poisson Distribution Continuous Probability Distributions The Uniform Distribution lecture 10, Discrete Random Variables III & Continuous Random Variable I Poisson Distribution Mean and Variance of a Poisson Random Variable If X is a Poisson ( λ ) random variable, then • Mean = λ • Variance = λ • Standard deviation = √ λ lecture 10, Discrete Random Variables III & Continuous Random Variable I 6/26 Poisson Distribution Continuous Probability Distributions The Uniform Distribution lecture 10, Discrete Random Variables III & Continuous Random Variable I...
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Lecture10 - lecture 10, Discrete Random Variables III...

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