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CIE A-Level Maths Study Notes

5.2.4 Poisson Distribution of Linear Combinations

Comprehending the role of the Poisson distribution in linear combinations is vital. This topic illuminates the intriguing property of independent Poisson-distributed random variables summing up to follow a Poisson distribution, a cornerstone concept in statistical analysis.

Understanding the Poisson Distribution

The Poisson distribution is used to model the probability of a number of events happening in a fixed interval of time or space, given a known constant mean rate of occurrence and the independence of events.

  • Key Traits of Poisson Distribution:
    • Independence of events.
    • Constant average rate (λ, lambda).
    • Suitable for low-frequency events over large intervals.

Linear Combinations with Poisson Distribution

A linear combination in statistics involves adding or subtracting variables, often multiplied by constants. For Poisson distributions, this translates to combining different Poisson processes.

Examples

Example 1: Combining Two Processes

Problem: Two independent Poisson processes have average rates of 3 and 5 events per hour. What is the distribution for the total events in an hour?

Solution:

  • Combine the Rates:
    • Process A(λA)A (λ_A) = 3, Process B(λB)B (λ_B) = 5.
    • Total rate λTotal=λA+λB=3+5=8.λ_Total = λ_A + λ_B = 3 + 5 = 8.
  • Poisson Distribution for Total Events:
    • The combined process follows a Poisson distribution with λ = 8.
Combining Two Processes Graph

Example 2: Real-World Call Center

Problem: A call centre gets Type A calls at a rate of 2 per hour and Type B calls at 4 per hour. Calculate the probability of exactly 5 calls in an hour.

Solution:

  • Calculate Combined Rate:
λ=λA+λB=2+4=6λ = λ_A + λ_B = 2 + 4 = 6
  • Probability for 5 Calls:
    • Use Poisson formula: P(X=5)=(e(6)65)/5!.P(X = 5) = (e^(-6) * 6^5) / 5!.
    • Perform the calculation to find P(X = 5).
Real-World Call Center Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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