Question

In the Poisson probability distribution, if the value of $\lambda$ is integer then the distribution will be :

• A. Bimodal
• B. Unimodal
• C. Positive modal
• D. Negative modal

Hint:

If \( \lambda \) is an integer, the two modes of the Poisson distribution are \( \lambda \) and \( \lambda - 1 \).
For example, if \( \lambda = 3 \), the modes are at \( X = 2 \) and \( X = 3 \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The mode of a discrete probability distribution is the value of the random variable at which the probability mass function achieves its maximum value.
Key Formula or Approach:
For a Poisson distribution with parameter \( \lambda \), the ratio of consecutive probabilities is:
\[ \frac{P(X = k)}{P(X = k-1)} = \frac{\lambda}{k} \]

Step 2: Detailed Explanation:
Let us analyze the behavior of this ratio to find the mode:
- If \( k < \lambda \), then \( \frac{P(X=k)}{P(X=k-1)} > 1 \implies P(X=k) > P(X=k-1) \), meaning the probabilities are strictly increasing.
- If \( k > \lambda \), then \( \frac{P(X=k)}{P(X=k-1)} < 1 \implies P(X=k) < P(X=k-1) \), meaning the probabilities are strictly decreasing.
Now, let us consider the case where the parameter \( \lambda \) is an integer:
Let \( k = \lambda \). The ratio becomes:
\[ \frac{P(X = \lambda)}{P(X = \lambda-1)} = \frac{\lambda}{\lambda} = 1 \implies P(X = \lambda) = P(X = \lambda - 1) \]
This means the maximum probability is achieved at two distinct consecutive values: \( X = \lambda - 1 \) and \( X = \lambda \).
Because there are two distinct modes, the Poisson distribution in this case is bimodal.
If \( \lambda \) were a non-integer, there would be a unique maximum at \( X = \lfloor \lambda \rfloor \), making it unimodal.

Step 3: Final Answer:
The distribution will be Bimodal.