1. STATISTICS “The Poisson Probability Distribution” 10.0 The Poisson Probability Distribution

2. Characteristics of a Poisson Process –A Poisson Process has the following characteristics: 1)The experiment consists of counting the number of occurrences of an event over a period of time, area, distance or any type of measurement 2)The mean of the Poisson distribution has be the same for each interval of measurement 3)The number of occurrences during one interval is independent of the number of occurrences in any other interval. 10.0 The Poisson Probability Distribution

3. Characteristics of a Poisson Process –Examples of Poisson Process: 1)The number of cars that arrive at a tollbooth over a specific period of time 2)The number of students who are absent in my Mondaystatistics class 3)The number of Man Utd players who are placed on the injured list each week 4)The number of cars using the Bangi toll each month 10.0 The Poisson Probability Distribution

4. The Poisson Probability Distribution –To make Poisson Probability Distribution, we need to know the average number of occurrences for the event over the interval. (mean) –If a random variable follows a pattern consistent with a Poisson probability distribution, we can calculate the probability of a certain number of occurrences over a given interval 10.0 The Poisson Probability Distribution

5. The Poisson Probability Distribution 1)We can calculate the Poisson Probability Distribution given the average over an interval, as follows: P[x] = μ. e x! x-μ-μ x = the number of occurrences of interest over the interval μ = the mean number of occurrences over the interval e = the mathematical constant 2.71828 10.0 The Poisson Probability Distribution

6. The Poisson Probability Distribution 2)The relationship between variance and the mean for a Poisson Probability Distribution is σ² = μ 3)The Standard deviation for a Poisson Probability Distribution can be calculated using the formula: σ = √ μ σ² = the variance of PPD μ = the mean of the PPD 10.0 The Poisson Probability Distribution

7. Useful term in Probability distribution: 1) “ Exactly ” & ” empty ” = no accumulative, single number only. Eg. What is the probability it will rain exactly 4 days in the next month. : P[x=4] 2) “ Less than ”,” Fewer than ”, = accumulative of lower value Eg. What is the probability it will rain less than 4 days in the next month. : P[x<4] = P[0]+P[1]+P[2]+P[3] 10.0 The Poisson Probability Distribution

8. Useful term in Probability distribution: 3) “ Not more than ”,” less or equal than ”, = accumulative and equal of lower value Eg. What is the probability it will rain not more than 4 days in the next month. : P[x ≤ 4] = P[0]+P[1]+P[2]+P[3]+P[4] 4) “More than” = accumulative of upper value Eg. What is the probability it will rain more than 4 days in the next month. : P[x>4] = P[5]+P[6]+P[7]+P[8]+P[9]……. = 1 – (P[0]+P[1]+P[2]+P[3]+P[4]) 10.0 The Poisson Probability Distribution

9. Useful term in Probability distribution: 5) “ At least ”,” more or equal ”, = accumulative and equal of upper value Eg. What is the probability it will rain at least 4 days in the next month. : P[x≥4] = P[4]+P[5]+P[6]+P[7]+P[8]+…… = 1 – (P[0]+P[1]+P[2]+P[3]) 10.0 The Poisson Probability Distribution

10. TRY THIS!!! 1)The number of rainy days per month at a particular town fallows a Poisson distribution with a mean value of 6 days. What is the that it will rain 4 days next month? (Answer: 0.1339) 2)The number of customers arriving at a particular store follows a Poisson Distribution with mean value of 4.2 customers per hour. What is the probability that more than 2 customers will arrive during the next hour? (Answer: 0.7897) 3) Suppose the average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than four lions on the next 1-day safari? (Answer: 0.2650 ) 10.0 The Poisson Probability Distribution

11. TRY THIS!!! 4) The average number of homes sold by the Acme Realty Company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow. (Answer: 0.180) 5) The number of spam email that Ronaldo receive each day follows a Poisson Distribution with a mean of 2.5 emails. What is the probability that he will receive more than 4 spam emails tomorrow? Std deviation? (Answer: 0.109; 1.58 emails) b) What is the probability that he will receive more than 1 but less than 5 accidents (Answer: 0.604 ) 10.0 The Poisson Probability Distribution

12. TRY THIS!!! During the year 2000, MAS had hired a number of employees from various graduate school. With the total average number of pilot hired is 7, use Poisson Probability Distribution to calculate: i)MAS hired 8 pilots next year ii)MAS hired fewer than 4 pilots next year iii)MAS hired not more than 6 pilots next year iv)MAS hired more than 2 pilots next year v)MAS hired at least 5 pilots next year vi)MAS hired more than 3 pilots and less than or equal 6 10.0 The Poisson Probability Distribution

13. During the year 2005, Air Asia had carried out a survey on flight delays. With the statistical data provided that year, an average of 4.55 flights were delayed. With the average value provided, use Poisson Probability Distribution to calculate: i)Air Asia has 6 flight delays next year ii)Air Asia has fewer than 5 flight delays next year iii)Air Asia has not more than 3 flight delays next year iv)Air Asia has more than 2 flight delays next year v)Air Asia has at least 4 flight delays next year vi)Air Asia has more than 1 flight delays and not more than 7 flight delays vii)The variance and Std deviation of the PPD QUIZ 7