Question

A random variable \( X \) follows a Poisson distribution with mean 5. Find the probability that \( X3 \).

• A. \( \frac{37}{2 e^5} \)
• B. \( 6 e^5 \)
• C. \( 6 e^{-5} \)
• D. \( \frac{37}{2 e^{-5}} \)

Hint:

For Poisson distributions, sum individual probabilities up to the desired value.

Answer 1

The Correct Option is D

Step 1: Poisson Probability Formula
\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \] Given \( \lambda = 5 \), we find: \[ P(X3) = P(0) + P(1) + P(2) \] \[ P(0) = \frac{e^{-5} 5^0}{0!} = e^{-5} \] \[ P(1) = \frac{e^{-5} 5^1}{1!} = 5 e^{-5} \] \[ P(2) = \frac{e^{-5} 5^2}{2!} = \frac{25}{2} e^{-5} \] Step 2: Summation
\[ P(X3) = e^{-5} + 5 e^{-5} + \frac{25}{2} e^{-5} \] \[ = \left(1 + 5 + \frac{25}{2} \right) e^{-5} \] \[ = \frac{37}{2} e^{-5} \] Thus, the correct answer is \( \frac{37}{2 e^{-5}} \).