Question

If a random variable \(X\) has a Poisson distribution with parameter \(\frac{1}{2}\), then \(P(X=2) = _ _ _\)

• A. \(\frac{1}{2\sqrt{e}}\)
• B. \(\frac{1}{4\sqrt{e}}\)
• C. \(\frac{1}{8\sqrt{e}}\)
• D. 1/2

Hint:

Write negative fractional exponents in the denominator as square roots: \(e^{-1/2} = \frac{1}{\sqrt{e}}\).
Keep track of fractions in the numerator of your probability calculation to avoid dividing errors at the final step.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the probability that a Poisson-distributed random variable \(X\) takes the value of 2, given that the distribution parameter (mean rate \(\lambda\)) is \(\frac{1}{2}\).

Step 2: Key Formula or Approach:
The probability mass function (PMF) of a Poisson distribution is given by the formula:
\[ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \] where:
\(\lambda\) is the parameter of the distribution (average rate of occurrence).
\(x\) is the actual number of occurrences (non-negative integer).
\(e\) is Euler's constant.

Step 3: Detailed Explanation:
Let us list the given parameters for this specific problem:
- Poisson parameter, \(\lambda = \frac{1}{2}\)
- Value of interest, \(x = 2\)
Substitute these values into the PMF formula:
\[ P(X = 2) = \frac{e^{-1/2} \left(\frac{1}{2}\right)^2}{2!} \] Let us calculate each term in the fraction step-by-step:
1. Calculate the numerator terms:
- Exponential term: \(e^{-1/2} = \frac{1}{e^{1/2}} = \frac{1}{\sqrt{e}}\)
- Polynomial term: \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
So the numerator is:
\[ e^{-1/2} \left(\frac{1}{2}\right)^2 = \frac{1}{\sqrt{e}} \times \frac{1}{4} = \frac{1}{4\sqrt{e}} \] 2. Calculate the denominator term:
- Factorial term: \(2! = 2 \times 1 = 2\)
3. Combine both parts to find the final probability:
\[ P(X = 2) = \frac{\frac{1}{4\sqrt{e}}}{2} = \frac{1}{4\sqrt{e} \times 2} = \frac{1}{8\sqrt{e}} \] This matches Option (C) exactly.

Step 4: Final Answer
Therefore, the probability \(P(X=2)\) is \(\frac{1}{8\sqrt{e}}\), which corresponds to option (C).