Learning Outcomes: Identify the Poisson distribution (a) *Cite some Poisson process *Highlight the situation of when to use Binomial distribution & Poisson distribution. (b) Find the mean and variance of Poisson distribution (c) Find the probability by using Poisson distribution At the end of the lesson, students should be able to Poisson Random Variable A Poisson Random Variable is a discrete random variable which represents the number of times a random event occurs in an interval of time or space ▪ The number of accidents occurring on a highway in a day ▪ The number of phone calls received from 8am – 9am ▪ The number of typing mistakes on a page ▪ The number of bacteria in of lake water E.g. Poisson Distribution, The Poisson Distribution, is a discrete probability distribution of a poisson random variable with a given mean rate, and its function given by : ▪ (NON-negative integers) ▪ (within a given interval of time & space) where Mean & Variance of Poisson Distribution Mean of Variance of For , 9.2 Poisson Distribution C09 -SM025 Page 1
The table of Binomial Distribution is constructed based on the cumulative function given by the formula: for Interpretation from the Statistical table Based on the table above, for . Verification by calculation using formula: Table of Poisson Distribution C09 -SM025 Page 2
Given . Find (a) Solution: (b) Solution: (c) Solution: (d) Solution: (e) Solution: (f) Solution: Example 1 C09 -SM025 Page 3
The number of telephone calls made to a switchboard during an afternoon can be modeled by a Poisson distribution with a mean of eight calls per five-minute. Find the probability that in the next five minutes : (a) no calls is made (b) five calls are made (c) at least three calls are made Solutions: Let per 5 minutes (a) (b) (c) Example 2 (a) (b) (c) C09 -SM025 Page 4
The number of telephone calls made to a switchboard during an afternoon can be modeled by a Poisson distribution with a mean of eight calls per five-minute. Find the probability that in the next five minutes : (d) less than four calls are made (e) i. between three and five calls are made ii. between three and five calls are made(inclusive). Solutions: Let per 5 minutes (d) (e) (i) (ii) Example 2 (d)(e) C09 -SM025 Page 5
The number of industrial injuries per working week in a particular factory is known to follow a Poisson distribution with mean 0.5. Find the probability that (i) less than 2 accidents, (ii) more than 2 accidents; (a) in a particular week there will be (b) there will be no accidents in a three week period . Solutions: Let (a) per week (i) (ii) (b) Remarks: [Changes in interval] 1 week period : 3 week period : Example 3 C09 -SM025 Page 6
If the random variable follows a Poisson distribution with mean 9, find (a) (b) (c) Solutions: (a) (b) (c) Example 4 C09 -SM025 Page 7
The number of phone calls received by a household each day over a period of 300 days was recorded and the results were as follows: Number of calls 0 1 2 3 4 Number of days 121 110 50 16 3 (a) Find the . Solution: (b) If follows a Poisson Distribution, state . ** Solution: Example 5 ** C09 -SM025 Page 8