LECTURE 18The Poisson Process: ReviewExample: Poisson CatchesExample: Poisson CatchesAdding (Merging) Poisson ProcessesSplitting of Poisson ProcessesExample: Email Filter (1)Example: Email Filter (2)Example: Email Filter (3)LECTURE 18• Readings: Finish Section 5.2Lecture outline• Review of the Poisson process• Properties– Adding Poisson Processes– Splitting Poisson Processes• ExamplesThe Poisson Process: Review• Number of arrivals in disjoint time intervals are independent, = “arrival rate”(for very small )(Poisson)• Interarrival times (k =1): (Exponential)• Time to the arrival: (Erlang)Example: Poisson Catches• Catching fish according to Poisson .• Fish for two hours, but if there’s no catch, continue until the first one.Example: Poisson Catches• Catching fish according to Poisson .• Fish for two hours, but if there’s no catch, continue until the first one.Adding (Merging) Poisson Processes• Sum of independent Poisson random variables is Poisson.• Sum of independent Poisson processes is Poisson.Red light flashesSome light flashesGreen light flashes• What is the probability that the next arrival comes from the first process?Splitting of Poisson Processes• Each message is routed along the first stream with probability , and along the second stream with probability .– Routing of different messages are independent.ServerUSAEmail traffic leaving MITForeign– Each output stream is Poisson.Example: Email Filter (1)• You have incoming email from two sources: valid email, and spam. We assume both to be Poisson. • Your receive, on average, 2 valid emails per hour, and 1 spam email every 5 hours.ValidIncoming EmailSpam• Total incoming email rate =• Probability that areceived email is spam =Example: Email Filter (2)• You install a spam filter, that filters out spam email correctly 80% of the time, but also identifies a valid email as spam 5% of the time. Spam FolderValidInboxSpam•• Inbox email rate =• Spam folder email rate =Example: Email Filter (3)Spam Folder• Probability that an emailin the inbox is spam = • Probability that an email in the spam folderis valid = ValidInboxSpam• Every how often should you check your spam folder, to find one valid email, on
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