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

The correct answer is 3 and 3, respectively.

Step-by-Step Explanation:

To find the mean and variance of the distribution, we can model the number of defective components using the Poisson approximation to the Binomial distribution. This approximation is highly accurate when the number of trials is large and the probability of a defect is small.

As shown in the provided image, we are given the following values:

The probability of producing a defective component:
$P=0.015$

The number of components in a packed box:
$n=200$

1. Finding the Mean:

For a Poisson distribution, the mean number of occurrences (denoted by the Greek letter lambda) is calculated as the product of the number of trials and the probability of success:

$\mathrm{Mean}=\lambda =n\times P$

Substituting the given values from the image:

$\lambda =200\times 0.015=3$

2. Finding the Variance:

A key mathematical property of the Poisson distribution is that its mean is exactly equal to its variance:

$\mathrm{Variance}={\sigma }^2=\lambda$

Since the mean is 3, the variance is also:

${\sigma }^2=3$

Thus, the mean and variance of the distribution are 3 and 3, respectively.