Question

Let \( X \) be a Poisson random variable and \( P(X = 1) + 2P(X = 0) = 12P(X = 2) \). Which one of the following statements is TRUE?

• A. \( 0.40<P(X = 0) \leq 0.45 \)
• B. \( 0.45<P(X = 0) \leq 0.50 \)
• C. \( 0.50<P(X = 0) \leq 0.55 \)
• D. \( 0.55<P(X = 0) \leq 0.60 \)

Hint:

In Poisson distributions, the probability of observing \( k \) events is given by the formula \( P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \). Use this formula to solve for \( \lambda \) when you have multiple conditions involving probabilities.

Answer 1

The Correct Option is C

Step 1: Express the Poisson probabilities.
For a Poisson distribution, the probability mass function is given by: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots \] We are given the equation \( P(X = 1) + 2P(X = 0) = 12P(X = 2) \). Substituting the Poisson PMF for each term, we get an equation involving \( \lambda \).
Step 2: Solve for \( \lambda \).
Solve the equation for \( \lambda \) by substituting the expressions for \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \).
Step 3: Analyze the options.
After solving for \( \lambda \), calculate \( P(X = 0) \) and determine which option matches the result.
Step 4: Conclusion.
The correct answer is (B) \( 0.45<P(X = 0) \leq 0.50 \).