LECTURE 16. Readings: Section 5.1. Lecture outline. Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process

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1 LECTURE 16 Readings: Section 5.1 Lecture outline Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process Number of successes Distribution of interarrival times The time of the success

2 Random Processes: Motivation Sequence of random variables: Examples: Arrival example: Arrival of people to a bank. Queuing example: Length of a line at a bank. Gambler s ruin: The probability of an outcome is a function of the probability of other outcomes (Markov Chains). Engineering example: Signal corrupted with noise.

3 The Bernoulli Process A sequence of independent Bernoulli trials. At each trial: T T T H T T T H H T T T T T H T H T T H $ $ $ $ $ $ Examples: Sequence of ups and downs of the Dow Jones. Sequence of lottery wins/losses. Arrivals (each second) to a bank.

4 Number of successes in time slots (Binomial) Mean: Variance:

5 Interarrival Times : number of trials until first success (inclusive). (Geometric) Memoryless property. Mean: Variance:

6 Fresh Start and Memoryless Properties Fresh Start Given n, the future sequence is a also a Bernoulli process and is independent of the past. Memorylessness Suppose we observe the process for n times and no success occurred. Then the pmf of the remaining time for arrival is geometric.

7 Time of the Arrival : number of trials until success (inclusive). : kth interarrival time It follows that:

8 Time of the Arrival : number of trials until success (inclusive). Mean: Variance: (Pascal)

9 LECTURE 17 Readings: Start Section 5.2 Lecture outline Review of the Bernoulli process Definition of the Poisson process Basic properties of the Poisson process Distribution of the number of arrivals Distribution of the interarrival time Distribution of the arrival time

10 The Bernoulli Process: Review Discrete time; success probability in each slot =. PMF of number of arrivals in time slots: Binomial PMF of interarrival time: PMF of time to arrival: Memorylessness Geometric Pascal What about continuous arrival times? Example: arrival to a bank.

11 The Poisson Process: Definition Let = Assumptions: Probability of arrivals in an interval of duration. Number of arrivals in disjoint time intervals are independent. For VERY small, we have: = arrival rate of the process.

12 From Bernoulli to Poisson (1) Bernoulli: Arrival prob. in each time slot = Poisson: Arrival probability in each -interval = Let and : Number of arrivals in a -interval = Number of successes in time slots (Binomial)

13 From Bernoulli to Poisson (2) Number of arrivals in a -interval as = (Binomial) (reorder terms) (Poisson)

14 PMF of Number of Arrivals : number of arrivals in a -interval, thus: (Poisson) Mean: Variance: Transform:

15 Example You get according to a Poisson process, at a rate of = 0.4 messages per hour. You check your every thirty minutes. Prob. of no new messages = Prob. of one new message =

16 Interarrival Time : time of the arrival. First order interarrival time: (Exponential) Why:

17 Interarrival Time Fresh Start Property: The time of the next arrival is independent from the past. Memoryless property: Suppose we observe the process for T seconds and no success occurred. Then the density of the remaining time for arrival is exponential. Example: You start checking your . How long will you wait, in average, until you receive your next ?

18 Time of Arrival : time of the arrival. : kth interarrival time It follows that:

19 Time of Arrival : time of the arrival. (Erlang) of order

20 Bernoulli vs. Poisson Times of Arrival Arrival Rate PMF of Number of Arrivals PMF of Interarrival Time PMF of Arrival Time Bernoulli Discrete /per trial Binomial Geometric Pascal Poisson Continuous /unit time Poisson Exponential Erlang

21 LECTURE 18 Readings: Finish Section 5.2 Lecture outline Review of the Poisson process Properties Adding Poisson Processes Splitting Poisson Processes Examples

22 The Poisson Process: Review Number of arrivals in disjoint time intervals are independent, = arrival rate (for very small ) (Poisson) Interarrival times (k =1): Time to the arrival: (Exponential) (Erlang)

23 Example: Poisson Catches Catching fish according to Poisson. Fish for two hours, but if there s no catch, continue until the first one.

24 Example: Poisson Catches Catching fish according to Poisson. Fish for two hours, but if there s no catch, continue until the first one.

25 Adding (Merging) Poisson Processes Sum of independent Poisson random variables is Poisson. Sum of independent Poisson processes is Poisson. Red light flashes Green light flashes Some light flashes What is the probability that the next arrival comes from the first process?

26 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. traffic leaving MIT Server USA Foreign Each output stream is Poisson.

27 Example: Filter (1) You have incoming from two sources: valid , and spam. We assume both to be Poisson. Your receive, on average, 2 valid s per hour, and 1 spam every 5 hours. Valid Spam Total incoming rate = Incoming Probability that a received is spam =

28 Example: Filter (2) You install a spam filter, that filters out spam correctly 80% of the time, but also identifies a valid as spam 5% of the time. Valid Spam Folder Inbox Spam Inbox rate = Spam folder rate =

29 Example: Filter (3) Valid Spam Folder Inbox Spam Probability that an in the inbox is spam = Probability that an in the spam folder is valid = Every how often should you check your spam folder, to find one valid , on average?

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