Question

The coefficient of variation of Poisson distribution with mean 36 is:

• A. \(\frac{1}{6} \times 100\)
• B. \(6 \times 100\)
• C. \(4 \times 100\)
• D. \(2 \times 100\)

Hint:

For any Poisson distribution, the Coefficient of Variation can be calculated directly as:
\[ \text{CV} = \frac{1}{\sqrt{\lambda}} \times 100 \]
Since \(\lambda = 36\), \(\sqrt{\lambda} = 6\), giving \(\frac{1}{6} \times 100\) immediately.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The Poisson distribution is a discrete probability distribution characterized by a single parameter, \(\lambda\), which represents both the mean and the variance of the distribution.
The coefficient of variation (CV) measures the relative dispersion of a distribution, expressed as a percentage of the mean.

Step 2: Key Formula or Approach:
For a Poisson distribution:
- \(\text{Mean} = \lambda\)
- \(\text{Variance} = \lambda \implies \text{Standard Deviation } (\sigma) = \sqrt{\lambda}\)
The formula for the Coefficient of Variation is:
\[ \text{CV} = \frac{\sigma}{\text{Mean}} \times 100 \]

Step 3: Detailed Explanation:
We are given:
- Mean of the Poisson distribution, \(\lambda = 36\)
Using the properties of the Poisson distribution, we find the variance:
\[ \text{Variance} = \lambda = 36 \]
Next, calculate the standard deviation \(\sigma\):
\[ \sigma = \sqrt{\text{Variance}} = \sqrt{36} = 6 \]
Now, substitute these values into our Coefficient of Variation formula:
\[ \text{CV} = \frac{\sigma}{\text{Mean}} \times 100 \]
\[ \text{CV} = \frac{6}{36} \times 100 \]
Simplify the fraction:
\[ \text{CV} = \frac{1}{6} \times 100 \]
This matches the expression in the first option.

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