Poisson lecture

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Poisson Lecture

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Poisson lecture

  1. 1. 1
  2. 2. The Poisson Experiment • Used to find the probability of a rare event • Randomly occurring in a specified interval – Time – Distance – Volume • Measure number of rare events occurred in the specified interval 2
  3. 3. The Poisson Experiment - Properties • Counting the number of times a success occur in an interval • Probability of success the same for all intervals of equal size • Number of successes in interval independent of number of successes in other intervals • Probability of success is proportional to the size of the interval • Intervals do not overlap 3
  4. 4. The Poisson Experiment - Properties • Some of the properties for a Poisson experiment can be hard to identify. • Generally only have to consider whether:- – The experiment is concerned with counting number of RARE events occur in a specified interval – These RARE events occur randomly – Intervals within which RARE event occurs are independent and do not overlap 4
  5. 5. The Poisson Experiment Typical cases where the Poisson experiment applies: – Accidents in a day in Soweto – Bacteria in a liter of water – Dents per square meter on the body of a car – Viruses on a computer per week – Complaints of mishandled baggage per 1000 passengers 5
  6. 6. The Poisson ExperimentCalculating the Poisson Probability Determining x successes in the interval:  e x P( X  x)  P( x)  x!where,  = mean number of successes in interval e = base of natural logarithm  2.71828 6
  7. 7. The Poisson Experiment - Example• On average the anti-virus program detects 2 viruses per week on a notebookAre the conditions required for the Poisson experiment met?• Time interval of one week• μ = 2 per week• Occurrence of viruses are independent• Can calculate the probabilities of a certain number of viruses in the interval 7
  8. 8. The Poisson Experiment - Example• Let X be the Poisson random variable indicating the number of viruses found on a notebook per weekCalculate the probability that threeviruses occur one viruses occur zerovirus occur  x e  P ( X  x)  P( x)  x! e2 20 P( X  0)  P(0)  0!  0.1353 e2 21 P( X  1)  P(1)  1!  0.2707 e2 23 P( X  3)  P(3)  3!  0.1804 e2 27 P( X  7)  P(7)   0.0034 8 . 7! .
  9. 9. The Poisson Experiment - Example• Let X be the Poisson random variable indicating the number of viruses found on a notebook per week X P(X) e2 20P( X  0)  P(0)  0!  0.1353 0 0.1353 1 0.2707 e2 21P( X  1)  P(1)  1!  0.2707 2 0.2707 3 0.1804 e2 23P( X  3)  P(3)  3!  0.1804 ↓ ↓ 7 0.0034 e2 27P( X  7)  P(7)   0.0034 ↓ ↓. 7!.. ∑P(X) ≈ 19
  10. 10. The Poisson Experiment - Example • Calculate the probability that two or less than two viruses will be found per week X P(X) P(X ≤ 2) 0 0.1353 1 0.2707 = P(X = 0) + P(X = 1) + P(X = 2) 2 0.2707 3 0.1804 = 0.1353 + 0.2707 + 0.2707 ↓ ↓ = 0.6767 7 0.0034 ↓ ↓ ∑P(X) ≈ 10 1
  11. 11. The Poisson Experiment - Example • Calculate the probability that less than three viruses will be found per week X P(X) P(X < 3) 0 0.1353 1 0.2707 = P(X = 0) + P(X = 1) + P(X = 2) 2 0.2707 3 0.1804 = 0.1353 + 0.2707 + 0.2707 ↓ ↓ = 0.6767 7 0.0034 ↓ ↓ ∑P(X) ≈ 11 1
  12. 12. The Poisson Experiment - Example • Calculate the probability that more than three viruses will be found per week X P(X) P(X > 3) 0 0.1353 1 0.2707 = P(X = 4) + P(X = 5) + ……… 2 0.2707 3 0.1804 = 1 – P(X ≤ 3) ↓ ↓ = 1 - 0.8571 7 0.0034 ↓ ↓ =0.1429 ∑P(X) ≈ 12 1
  13. 13. The Poisson Experiment - Example • Calculate the probability that four viruses will be found in four weeks • μ = 2 x 4 = 8 in four weeks P ( X  4) e 8 84  4!  0.0573 13
  14. 14. The Poisson Experiment - Example • Calculate the probability that two or less than two viruses will be found in two weeks • μ = 2 x 2 = 4 in two weeks P( X  2)  P( X  0)  P( X  1)  P( X  2) 4 0 4 1 4 2 e 4 e 4 e 4    0! 1! 2!  0.0183  0.0733  0.2381  0.2381 14
  15. 15. The Poisson Experiment • Mean and standard deviation of Poisson random variable   E( X )       Var ( X )   2 15
  16. 16. The Poisson Experiment - Example – What is the expected number of viruses on the notebook per week?   E( X )    2 – What is the standard deviation for the number of viruses on the notebook per week?     Var ( X )    2  1.41 2 16

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