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The Poisson Distribution is used when we have the average number of times an event occurs.

The Poisson Distribution. The Poisson Distribution is used when we have the average number of times an event occurs. e.g. the number of phone calls received in an office in 1 day, the number of flaws along a specified length of material,

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The Poisson Distribution is used when we have the average number of times an event occurs.

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  1. The Poisson Distribution The Poisson Distribution is used when we have the average number of times an event occurs. e.g. the number of phone calls received in an office in 1 day, the number of flaws along a specified length of material, the mean number of accidents along a specific stretch of the A470 in a month.

  2. If X has a Poisson distribution, we write X ~ Po ( μ ) μ = the mean P(X = x) = e-µ µ x x! It is also possible to use tables to find the probability for the Poisson distribution.

  3. The Mean and Variance of the Poisson Distribution The mean is given to us as µ and the variance is always equal to the mean in a Poisson distribution. E(X) = Var(X) = µ Remember that the standard deviation is the square root of the variance. Standard Deviation = √Var(X) = √µ

  4. Example • X has the Poisson distribution with mean 5. Find • P(X = 4) • P(X ≥ 6) • P(2 ≤ X ≤ 4) • P(X < 2) • mean & variance of Y when Y = 4X - 2

  5. X ~ Po ( 5 ) = 0.175 = e-µ µ x x! = e-5 54 4! a) P(X = 4) = 0.384 (using tables) b) P(X ≥ 6) = P(X ≥ 2) - P(X ≥ 5) = 0.9596 – 0.5595 c) P(2 ≤ X ≤ 4) = 0.4001 d) P(X < 2) = P(X = 0) + P(X = 1) = e-5 50 + e-5 51 0!1! = 0.00674 + 0.0337 = 0.0404 = 4E(X) - 2 = 18 e) E(Y) = E(4X – 2) = 4 x 5 - 2 Var(Y) = Var(4X – 2) = 42 Var(X) =16 x 5 = 80

  6. Exercise 4.6a 4.6b 4.6c Mathematics Statistics Unit S1 - WJEC

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