Question

The mean and variance of Poisson distribution are both:

• A. 0
• B. 1
• C. \(\lambda\)
• D. \(1/\lambda\)

Hint:

A unique feature of the Poisson distribution is that its Mean = Variance = \(\lambda\). This property of equidispersion is often used to test if a given spatial distribution of insects fits a Poisson model.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The Poisson distribution is a discrete probability distribution that models the number of times an event occurs in a fixed interval of time or space.

Step 2: Key Formula or Approach:
The probability mass function of a Poisson-distributed random variable \(X\) is given by:
\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]
where \(\lambda\) is the parameter representing the average rate of occurrence.

Step 3: Detailed Explanation:
For a Poisson distribution, the statistical properties are such that:
\[ \text{Mean } (\mu) = E(X) = \lambda \]
\[ \text{Variance } (\sigma^2) = Var(X) = \lambda \]
This equality of mean and variance (\(\mu = \sigma^2 = \lambda\)) is a unique and defining characteristic of the Poisson distribution.

Step 4: Final Answer:
Both mean and variance are equal to \(\lambda\), which corresponds to option (C).