S2 Poisson Distribution.

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Presentation transcript:

S2 Poisson Distribution

The conditions required for a Poisson distribution are: Events occur at random Events occur independently of each other The average rate of occurrences remains constant There is zero probability of simultaneous occurrences The Poisson distribution is defined as… 𝑿 ~ 𝑷𝒐(𝝀) 𝑷 𝑿=𝒓 = 𝒆 −𝝀 𝝀 𝒓 𝒓! 𝒇𝒐𝒓 𝒓=𝟎, 𝟏, 𝟐, 𝟑,… λ is the only parameter and represents the mean number of occurrences in the time period.

If X ~ Po(3), find P(X = 2)

The number of cars passing a point on a road in a 5-minute Period may be modelled by a Poisson distribution with Parameter 4. Find the probability that, in a 5-minute period (a) 2 cars go past (b) fewer than 3 cars go past

The number of accidents in a week on a stretch of road is known to follow a Poisson distribution with parameter 2.1. Find the probability that In a given week there is 1 accident In a two week period there are 2 accidents There is 1 accident in each of two successive weeks.

The number of flaws in a metre length of dress material is known to follow a Poisson distribution with parameter 0.4. Find the probabilities that: There are no flaws in a 1-metre length There is 1 flaw in a 3-metre length There is 1 flaw in a half-metre length.

USING TABLES If X ~ Po(8.5) find: (a) P(X < 7) (b) P(X ≥ 9) (c) P(X > 4) (d) P(4 < X < 9)

MEAN AND VARIANCE OF THE POISSON DISTRIBUTION If 𝑿 ~ 𝑷𝒐(𝝀) then 𝑬 𝑿 =𝝀 𝑽𝒂𝒓 𝑿 =𝝀 The number of calls arriving at a switchboard in a 10-minute period can be modelled by a Poisson distribution with parameter 3.5. Give the mean and variance of the number of calls which arrive in (a) 10 minutes (b) an hour (c) 5 minutes

A dual carriageway has one lane blocked off due to roadworks A dual carriageway has one lane blocked off due to roadworks. The number of cars passing a point in a road in a number of one-minute intervals is summarised in the table. Calculate the mean and variance of the number of cars passing in one minute intervals. Is the Poisson distribution likely to be an adequate model for the distribution of the number of cars passing in one-minute intervals? Number of Cars 1 2 3 4 5 6 Frequency 25 30

The number of cyclists passing a village post office during the day can be modelled as a Poisson random variable. On average two cyclists pass by in an hour. What is the probability that Between 10am and 11am (i) no cyclists passes (ii) more than 3 cyclists pass Exactly one cyclist passes while the shop-keeper is on a 20-minute tea break. More than 3 cyclists pass in an hour exactly once in a six-hour period?

APPROXIMATING BINOMIAL WITH POISSON If X ~ B(n, p) and n is large (>50) and p is small (<0.1) Then you can approximate X using the Poisson distribution where λ = np 𝑿 ~ 𝑩 𝒏, 𝒑 ≈ 𝒀 ~ 𝑷𝒐(𝒏𝒑)

The probability that a component coming off a production line is faulty is 0.01. If a sample of size 5 is taken, find the probability that one of the components is faulty. What is the probability that a batch of 250 of these components has more than 3 faulty components in it?