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Unformatted text preview: 1 STAT 1301 Probability & Statistics Lecture 14 Lecture 14 Exponential and Gamma Distributions Introduction • In many applications one needs to model the random time of some kind. Examples are: – The time it takes to service a customer – The lifetime of a component – The time a patient survives after an operation • Such a random time T can be regarded as a random variable with distribution defined by some pdf f(t) with t in [0, ). So • The survival function is defined as . ) ( ( b a dt t f b) T a P . ) ( ( s dt t f s) T P Exponential Distribution • The simplest model for a random time with no upper bound on its range is the exponential distribution. • A random time T has an exponential distribution if the pdf of T is f ( t; ) = e- t , if t 0. = 0, if t < 0. Mean & Variance • Mean & variance: = E(T) = 1/ , 2 = V(T) = 1 2 = V(T) = 1/ . 2 cdf • The cdf of an exponential distribution can be easily calculated. So survival function . 1 ) ; ( t e t F t • So, survival function • The exponential distribution is often used as a model for the distribution of waiting times. The distribution is closely related to the Poisson process. . , ) ( t e t T P t Poisson Process • Counts of arrivals: The distribution of the number of arrivals N(t) in a fixed time interval of length t is Poisson ( t ) , and the number of arrivals in disjoint time intervals are independent. • Times between arrivals: The distribution of the waiting time W 1 until the first arrival is Exponential( ) , and W 1 and the subsequent waiting times W 2 , W 3 , … between each arrival...
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