Question

For a Poisson distribution where the mean is \(4\), what is the value of the third central moment?

• A. \(4\)
• B. \(8\)
• C. \(16\)
• D. \(64\)

Hint:

For a Poisson distribution, several moments have simple forms: \[ \text{Mean} = \text{Variance} = \text{Third central moment} = \lambda \] This property makes calculations involving Poisson moments very straightforward.

Answer 1

The Correct Option is A

Concept:
For a Poisson distribution with parameter \( \lambda \):

• Mean \(= \lambda\)
• Variance \(= \lambda\)
• The third central moment \( \mu_3 \) is also equal to \( \lambda \)

Thus, \[ \mu_3 = \lambda \]
Step 1: Identify the parameter of the distribution.
The mean of the Poisson distribution is given as: \[ \lambda = 4 \]
Step 2: Use the formula for the third central moment.
\[ \mu_3 = \lambda \] Substituting the value: \[ \mu_3 = 4 \] \[ \therefore \text{The third central moment is } 4 \]