Question

Let a random variable \( X \) follow Poisson distribution such that
\[ \text{Prob}(X = 1) = \text{Prob}(X = 2). \] The value of \( \text{Prob}(X = 3) \) is ________________ (round off to 2 decimal places).

Hint:

To solve for the Poisson probability when given conditions like \( P(X = 1) = P(X = 2) \), use the Poisson distribution formula and solve for \( \lambda \).

Answer 1

Correct Answer: 0.17

For a Poisson distribution, the probability mass function is given by:
\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where \( \lambda \) is the mean of the distribution, and \( k \) is the value for which we want to find the probability.
Given that \( P(X = 1) = P(X = 2) \), we can set up the equation: \[ \frac{\lambda^1 e^{-\lambda}}{1!} = \frac{\lambda^2 e^{-\lambda}}{2!} \] Simplifying the equation: \[ \lambda = \frac{\lambda^2}{2} \] Solving for \( \lambda \): \[ \lambda = 2. \] Now that we have \( \lambda = 2 \), we can calculate \( P(X = 3) \) using the Poisson distribution formula:
\[ P(X = 3) = \frac{2^3 e^{-2}}{3!} = \frac{8 e^{-2}}{6}. \] Approximating \( e^{-2} \approx 0.1353 \), we get: \[ P(X = 3) = \frac{8 \times 0.1353}{6} \approx 0.1804. \] Thus, the value of \( \text{Prob}(X = 3) \) is \(0.18 \).