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CIE A-Level Maths Study Notes

5.1.2 Mean and Variance of the Poisson Distribution

Understanding the Mean and Variance of the Poisson Distribution is a vital aspect. This detailed exploration focuses on the practical applications and implications of these concepts.

Overview of the Poisson Distribution

  • What is it? A probability distribution used to model the number of events in a fixed time or space interval.
  • Key Feature: Mean λλ = Variance λλ.
  • Mean μμ: The expected number of events, equal to λ.
  • Variance σ2σ²: The spread of the distribution, also equal to λ.

Applications

1. Predicting Event Counts:

  • High λ: More events, greater spread.
  • Low λ: Fewer events, lesser spread.

Examples

Example 1: Library Book Returns

Calculate and visualize the mean and variance of a library that averages 4 book returns per hour (λ = 4).

Solution:

1. Mean (μ) of the Distribution:

  • For a Poisson distribution, the mean (μ) is equal to the rate (λ) of the event.
  • In this scenario, λ = 4 (4 book returns per hour).
  • Therefore, μ = 4.

2. Variance (σ²) of the Distribution:

  • The variance (σ²) in a Poisson distribution is also equal to λ.
  • Hence, σ² = λ = 4.
  • This means the average squared deviation from the mean number of book returns is also 4.

Answers:

  • Mean and Variance: Both are 4.
  • Graph: Shows the probability of different return counts in an hour.
Poisson Distribution Graph

Example 2: Email Receipts Probability

A website averages 2 emails per hour (λ = 2). Calculate the probability of receiving exactly 3 emails in an hour and visualize it.

Solution:

  • Formula: P(X=k)=eλλkk!P(X = k) = \frac{e^{-λ} λ^k}{k!} for k=3k = 3.
  • Calculating for ( k = 3 ):
    • Find P(X=3)P(X = 3) given λ=2.λ = 2.
    • Plugging in the values:
    • P(X=3)=e2233!P(X = 3) = \frac{e^{-2} 2^3}{3!}
    • P(X=3)=e2×86P(X = 3) = \frac{e^{-2} \times 8}{6} (since 3!3! equals 6)
    • P(X=3)=0.1804P(X=3) = 0.1804

Therefore, the probability P(X=3)P(X = 3) is about 0.1804, or 18.04%. This means there's an 18.04% chance of receiving exactly 3 emails in any given hour.

  • Graph: Visualizes the probability of receiving different email counts in an hour.
Poisson Distribution Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
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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