Question

If \( X \) is a Poisson variate such that \( P(X = 1) = P(X = 2) \), then \( P(X = 4) \) is equal to:

• A. \( \frac{1}{2e^2} \)
• B. \( \frac{1}{3e^2} \)
• C. \( \frac{2}{3e^2} \)
• D. \( \frac{1}{2e} \)

Hint:

In Poisson distribution, the probability of each event is determined by the rate parameter \( \lambda \).

Answer 1

The Correct Option is C

Step 1: Use the Poisson distribution.
We use the fact that for a Poisson distribution, the probability \( P(X = k) \) is given by: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] Given that \( P(X = 1) = P(X = 2) \), we solve for \( \lambda \) and use it to find \( P(X = 4) \).
Step 2: Conclusion.
Thus, \( P(X = 4) = \frac{2}{3e^2} \).
Final Answer: \[ \boxed{\frac{2}{3e^2}} \]