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 $λ$ = Variance $λ$.
• Mean $μ$: The expected number of events, equal to λ.
• Variance $σ²$: 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.

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

Therefore, the probability $P(X = 3)$ 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.