Question

A machine produces a defective component with a probability of 0.015. The number of defective components in a packed box containing 200 components produced by the machine follows a Poisson distribution. The mean and the variance of the distribution are

• A. 3 and 3, respectively
• B. \(\sqrt{3}\) and \(\sqrt{3}\), respectively
• C. 0.015 and 0.015, respectively
• D. 3 and 9, respectively

Hint:

A fundamental property to remember for the Poisson distribution is that the mean is always equal to the variance (\(\mu = \sigma^2 = \lambda\)). If a question states a process follows a Poisson distribution and asks for both mean and variance, you know they must be the same value. This can help you quickly eliminate options like (D) where the mean and variance are different.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The problem describes a scenario that follows a Poisson distribution. The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space. A key characteristic of the Poisson distribution is that its mean and variance are equal.
Step 2: Key Formula or Approach:
This situation can be viewed as a binomial process (each component is either defective or not) that is approximated by a Poisson distribution because the number of trials (\(n\)) is large and the probability of a defect (\(p\)) is small.
The parameter of the Poisson distribution, \(\lambda\), which represents both the mean and the variance, is calculated as:
\[ \lambda = n \times p \] For a Poisson distribution:
\[ \text{Mean} = \mu = \lambda \] \[ \text{Variance} = \sigma^2 = \lambda \] Step 3: Detailed Calculation:
Given:
- Number of components (trials), \(n = 200\)
- Probability of a defective component, \(p = 0.015\)
First, calculate the parameter \(\lambda\):
\[ \lambda = n \times p = 200 \times 0.015 \] \[ \lambda = 3.0 \] Now, according to the properties of the Poisson distribution:
\[ \text{Mean} = \lambda = 3 \] \[ \text{Variance} = \lambda = 3 \] Step 4: Final Answer:
The mean and the variance of the distribution are 3 and 3, respectively.
Step 5: Why This is Correct:
The calculation correctly uses the formula \(\lambda = np\) to find the parameter of the Poisson distribution. For any Poisson distribution, the mean is equal to the variance, and both are equal to \(\lambda\). Thus, both the mean and the variance are 3.