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

Step 1: Understanding the Question
The question asks for the value of the third central moment (\(\mu_3\)) for a Poisson distribution with a given mean.
Step 2: Key Formula or Approach
A key property of the Poisson distribution with parameter \(\lambda\) is that its mean, variance, and third central moment are all equal to \(\lambda\).

Mean \( E(X) = \lambda \)

Variance \( \text{Var}(X) = \mu_2 = \lambda \)

Third Central Moment \( \mu_3 = \lambda \)

Step 3: Detailed Explanation
The problem states that the mean of the Poisson distribution is 4.
From the properties of the Poisson distribution, we know that the mean is equal to the parameter \(\lambda\).
\[ \text{Mean} = \lambda = 4 \] The third central moment, \(\mu_3\), is also equal to \(\lambda\).
\[ \mu_3 = \lambda = 4 \] Step 4: Final Answer
The value of the third central moment is 4.