Question

For a Poisson Distribution, if mean(m) = 1, then P(1) is?

• A. 1.0
• B. 1/e
• C. e
• D. e/2

Hint:

For a Poisson distribution with mean $m = 1$, the probability of getting $0$ events and $1$ event is identical: $P(0) = P(1) = 1/e$. This is a unique property of the Poisson distribution when $m=1$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The Poisson distribution is a discrete probability distribution that models the number of events occurring within a fixed interval of time or space.
Key Formula or Approach:
The probability mass function of a Poisson distribution with mean parameter $m$ is: \[ P(X = k) = \frac{e^{-m} m^k}{k!} \]

Step 2: Detailed Explanation:
Given that the mean parameter $m = 1$:
We need to calculate the probability of getting exactly one event, $P(X = 1)$: \[ P(X = 1) = \frac{e^{-1} (1)^1}{1!} \]
Simplify the expression: \[ e^{-1} = \frac{1}{e} \] \[ 1^1 = 1 \] \[ 1! = 1 \]
Substituting these values back: \[ P(X = 1) = \frac{\left(\frac{1}{e}\right) \times 1}{1} = \frac{1}{e} \]

Step 3: Final Answer:
The value of $P(1)$ is 1/e.