Poissonprocesspre-reading

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School
University of British Columbia **We aren't endorsed by this school
Course
STAT 302
Subject
Statistics
Date
Aug 17, 2023
Pages
5
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STAT 302 In-class activity Poisson Process - Relationship between Poisson and Exponential Random variables Objectives : - Pre-reading: o Review the properties of an event that occurs according to a Poisson process o Recall that the number of occurrences of an event according to a Poisson process is a Poisson random variable o Recall that the time between two consecutive events that occur according to a Poisson process is an Exponential random variable o Compare graphically the difference between probability mass functions of the Poisson distributions with different mean values o Compare graphically the difference between cumulative distribution functions of the Exponential distributions with different rate parameters - In-class demonstration: o Examine the distribution of the times between earthquake occurrences of a real earthquake sequence by means of a histogram o Estimate average time between consecutive earthquakes in the earthquake sequence o Estimate the rate parameter of an Exponential distribution which is used to model the times between earthquake occurrences o Compare the histogram of the times between earthquake occurrences with the probability density function of the Exponential distribution with the estimated rate parameter - In-class activity: o Compare the shape of histogram of data of inter-arrival times of earthquakes with that of an Exponential distribution with a rate parameter estimated by the data o Calculate probabilities concerning the inter-arrival times of earthquakes based on the fitted Exponential model with the estimated rate parameter o Compare the theoretical probabilities calculated from the fitted Exponential model with the corresponding observed proportions in the data o Calculate probabilities concerning the number of occurrences of earthquakes over fixed time intervals based on the fitted Poisson model with the estimated rate parameter o Investigate the relationship between the Poisson distribution and the Exponential distribution
Pre-reading A Poisson distribution is a discrete probability distribution which gives the probability of having a certain number of events occurring in a fixed time period. The number of events per unit time can be assumed to follow a Poisson process with rate g2019 if the events occur one at a time, independently of each other and at random, past events do not influence future events (events have no "memory" of previous event), and the probability of having exactly one occurrence of the event on any time period of length g1872 is approximately g2019g1872 for small g1872 . These assumptions are valid or nearly so in a wide variety of natural and industrial systems. For a Poisson process with rate g2019 per unit time, the probability of having g1876 events in time g1872 is given by the probability mass function: g1868g4666g1876g4667 = g1842g4666g1850 = g1876g4667 = g1857 g2879g3090g3047 g4666g2019g1872g4667 g3051 g1876! g46661g4667 where g1850 is the Poisson random variable for the number of occurrences of the event. The following are the probability mass functions of the Poisson random variables with mean event rates of 2 and 5 per unit time. Notice that the distribution is right skewed (with a long right tail) with most of the probability mass concentrated about g2019g1872 . The distribution is more skewed for g2019 = 2 .
If the events follow a Poisson process with rate g2019 , the times between two consecutive events, g1846 , has an Exponential distribution. This can be shown using Equation (1), the probability mass function for the Poisson distribution. If no events occur in the fixed time period [ 0,g1872 ], then this is equivalent to writing that the time until the next event, g1846 , is greater than g1872 : g1842g4666g1846 > g1872g4667 = g1842g4666g1866g1867 g1867g1855g1855g1873g1870g1870g1857g1866g1855g1857g1871 g1867g1858 g1872ℎg1857 g1857g1874g1857g1866g1872 g1861g1866 g1872g1861g1865g1857 g1872g4667 = g1842g4666g1850 = 0g4667 = g1857 g2879g3090g3047 Therefore, g1842g4666g1846 ≤ g1872g4667 = 1 − g1842g4666g1846 > g1872g4667 = 1 − g1857 g2879g3090g3047 , which is the cumulative distribution function of the Exponential distribution. The probability density function for the Exponential distribution is g1858g4666g1872g4667 = g2019g1857 g2879g3090g3047 , g1872 > 0 The mean of the inter-arrival time of the events is g1831g4666g1846g4667 = g2869 g3090 . The variance is g1848g4666g1846g4667 = g2869 g3090 g3118 . The Exponential cumulative distribution functions (CDF) for different values of g2019 are shown below. Recall g1832g4666g1872g4667 = g1842g4666g1846 ≤ g1872g4667 = 1 − g1857 g2879g3090g3047 .
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