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# Poision distribution

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### Poision distribution

1. 1. POISION DISTRIBUTION Presented By:- 1. Shubham Ranjan 2. Siddharth Anand
2. 2. Introduction The distribution was first introduced by Simon Denis Poisson (1781–1840) and published, together with his probability theory, in 1837 in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile (“Research on the Probability of Judgments in Criminal and Civil Matters”).
3. 3. Definition The Poisson distribution is a probability model which can be used to find the probability of a single event occurring a given number of times in an interval of (usually) time. The occurrence of these events must be determined by chance alone which implies that information about the occurrence of any one event cannot be used to predict the occurrence of any other event.
4. 4. The Poisson Probability If X is the random variable then ‘number of occurrences in a given interval ’for which the average rate of occurrence is λ then, according to the Poisson model, the probability of r occurrences in that interval is given by P(X = r) = e−λλr /r ! Where r = 0, 1, 2, 3, . . . NOTE : e is a mathematical constant. e=2.718282 and λ is the parameter of the distribution. We say X follows a Poisson distribution with parameter λ.
5. 5. The Distribution arise When the Event being Counted occur • Independently • Probability such that two or more event occur simultaneously is zero • Randomly in time and space • Uniformly (no. of event is directly proportional to length of interval).
6. 6. Poisson Process Poisson process is a random process which counts the number of events and the time that these events occur in a given time interval. The time between each pair of consecutive events has an exponential distribution with parameter λ and each of these inter-arrival times is assumed to be independent of other inter-arrival times.
7. 7. Types of Poisson Process • Homogeneous • Non-homogeneous • Spatial • Space-time
8. 8. Example 1. Births in a hospital occur randomly at an average rate of 1.8 births per hour. What is the probability of observing 4 births in a given hour at the hospital? 2. If the random variable X follows a Poisson distribution with mean 3.4 find P(X=6)?
9. 9. The Shape of Poisson Distribution • Unimodal • Exhibit positive skew (that decreases a λ increases) • Centered roughly on λ • The variance (spread) increases as λ increases
10. 10. Mean and Variance for the Poisson Distribution • It’s easy to show that for this distribution, The Mean is: • Also, it’s easy to show that The Variance is: So, The Standard Deviation is:    2  
11. 11. Properties • The mean and variance are both equal to . • The sum of independent Poisson variables is a further Poisson variable with mean equal to the sum of the individual means. • The Poisson distribution provides an approximation for the Binomial distribution.
12. 12. Sum of two Poisson variables Now suppose we know that in hospital A births occur randomly at an average rate of 2.3 births per hour and in hospital B births occur randomly at an average rate of 3.1 births per hour. What is the probability that we observe 7 births in total from the two hospitals in a given 1 hour period?
13. 13. Comparison of Binomial & Poisson Distributions with Mean μ = 1 0 0.1 0.2 0.3 0.4 0.5 Probability 0 1 2 3 4 5m poisson binomial  N=3, p=1/3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Probability 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 m binomial poisson  N=10,p=0.1 Clearly, there is not much difference between them! For N Large & m Fixed: Binomial  Poisson
14. 14. Approximation If n is large and p is small, then the Binomial distribution with parameters n and p is well approximated by the Poisson distribution with parameter np, i.e. by the Poisson distribution with the same mean
15. 15. Example • Binomial situation, n= 100, p=0.075 • Calculate the probability of fewer than 10 successes. pbinom(9,100,0.075)[1] 0.7832687 This would have been very tricky with manual calculation as the factorials are very large and the probabilities very small
16. 16. • The Poisson approximation to the Binomial states that  will be equal to np, i.e. 100 x 0.075 • so =7.5 • ppois(9,7.5)[1] 0.7764076 • So it is correct to 2 decimal places. Manually, this would have been much simpler to do than the Binomial.
17. 17. THANK YOU