Question

Find the mean of x, if x is a Poisson variate satisfying the condition P(3)=P(4)

• A. 2
• B. 4
• C. 8
• D. 16

Hint:

For any Poisson distribution, a general relation holds:
\[ P(k) = P(k-1) \times \frac{\lambda}{k} \] If we set \( P(k) = P(k-1) \), we immediately get \(\lambda = k\). Here, \(P(4) = P(3)\) means \(\lambda = 4\). This rule allows you to solve the question mentally without writing down any calculations.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The Poisson distribution is a discrete probability distribution that models the number of times an event occurs in a fixed interval of time or space.
The single parameter of this distribution is \(\lambda\), which represents both the mean and the variance of the random variable.
Key Formula or Approach:
The probability mass function (PMF) of a Poisson distribution is given by:
\[ P(x; \lambda) = \frac{e^{-\lambda} \lambda^x}{x!}, \quad \text{for } x = 0, 1, 2, \dots \] Where: - \(\lambda\) is the mean of the distribution.
- \(x!\) is the factorial of \(x\).

Step 2: Detailed Explanation:
Let us use the given boundary condition \(P(3) = P(4)\) to solve for \(\lambda\):
1. Express \(P(3)\) and \(P(4)\) using the PMF formula:
\[ P(3) = \frac{e^{-\lambda} \lambda^3}{3!} \] \[ P(4) = \frac{e^{-\lambda} \lambda^4}{4!} \] 2. Equate the two probabilities:
\[ \frac{e^{-\lambda} \lambda^3}{3!} = \frac{e^{-\lambda} \lambda^4}{4!} \] 3. Since \(e^{-\lambda} \neq 0\) and \(\lambda > 0\), we can divide both sides of the equation by \(e^{-\lambda}\):
\[ \frac{\lambda^3}{3!} = \frac{\lambda^4}{4!} \] 4. Divide both sides by \(\lambda^3\):
\[ \frac{1}{3!} = \frac{\lambda}{4!} \] 5. Simplify the factorials:
We know that \(4! = 4 \times 3!\).
Substituting this:
\[ \frac{1}{3!} = \frac{\lambda}{4 \times 3!} \] 6. Multiply both sides by \(3!\) to isolate \(\lambda\):
\[ 1 = \frac{\lambda}{4} \implies \lambda = 4 \] The parameter \(\lambda\), which represents the mean of the Poisson variate \(x\), is 4.

Step 3: Final Answer:
The correct option is (B).