Question

If \(X\) is a random Poisson variate with \(P(X=1) = P(X=2)\). Then \(P(X=4)\) is:

• A. \(\frac{2e^{-2}}{3}\)
• B. \(\frac{e^{-2}}{3}\)
• C. \(\frac{4e^{-2}}{3}\)
• D. \(\frac{2e^{-3}}{6}\)

Hint:

For any Poisson distribution, if \(P(X = k) = P(X = k+1)\), then the parameter \(\lambda\) is simply equal to \(k+1\).
Here, \(P(X = 1) = P(X = 2) \implies \lambda = 1 + 1 = 2\). This saves time on algebra.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space.
The distribution is characterized by a single parameter, \(\lambda\) (lambda), which represents both the mean and the variance of the distribution.

Step 2: Key Formula or Approach:
The Probability Mass Function (PMF) of a Poisson-distributed random variable \(X\) is given by:
\[ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}, \quad \text{for } x = 0, 1, 2, \dots \]

Step 3: Detailed Explanation:
We are given that the probability of getting exactly 1 success is equal to the probability of getting exactly 2 successes:
\[ P(X = 1) = P(X = 2) \]
Using the Poisson PMF, we substitute \(x = 1\) and \(x = 2\) into the equation:
\[ \frac{e^{-\lambda} \lambda^1}{1!} = \frac{e^{-\lambda} \lambda^2}{2!} \]
Since the exponential term \(e^{-\lambda}\) is never zero for any real value of \(\lambda\), we can divide both sides of the equation by \(e^{-\lambda}\):
\[ \lambda = \frac{\lambda^2}{2} \]
Since \(\lambda\) must be positive for a non-trivial Poisson process (\(\lambda > 0\)), we can divide both sides by \(\lambda\):
\[ 1 = \frac{\lambda}{2} \implies \lambda = 2 \]
Thus, the mean parameter of the Poisson distribution is \(\lambda = 2\).
Now, we find the probability of the event \(X = 4\) using the value \(\lambda = 2\):
\[ P(X = 4) = \frac{e^{-2} 2^4}{4!} \]
We compute the numerical values:
\[ 2^4 = 16 \]
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
Substitute these values back into the probability formula:
\[ P(X = 4) = \frac{e^{-2} \times 16}{24} \]
Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, 8, we obtain:
\[ P(X = 4) = \frac{2 e^{-2}}{3} \]
This matches the first option.

Step 4: Final Answer:
Therefore, the correct option is (A).