91Ó°ÊÓ

In statistics, understanding when variables are independent is key to analyzing data accurately. In the context of the Poisson distribution, the errors on each page produced by the printer are considered independent random variables. This is because the occurrence of errors on one page does not influence the occurrence on another page. The Poisson process assumes that events happen independently over time or space.
For example, if a printer generates errors following a Poisson distribution, each page it prints is a separate event. The probability of errors on page one does not change the probability of errors on page two. This independent behavior is crucial for using the Poisson distribution model to predict and understand error occurrences in large volumes of printed pages.
Therefore, knowing that these errors are independent helps us apply the Poisson distribution accurately to determine the number of pages likely to have errors.

The mean of a Poisson distribution is the expected number of occurrences in a fixed interval, such as time or space. In this problem, we were given that the mean rate of errors is 0.4 per page. From here, we can calculate the expected number of pages that contain errors.
First, we find the probability of a single page having at least one error. The probability of zero errors on a page is calculated as follows:

• \[ P(X = 0) = e^{-0.4} \]

The complementary probability gives us the likelihood of having one or more errors:

• \[ P(X \geq 1) = 1 - e^{-0.4} \]

Since we have 1000 pages, the mean number of pages with errors can be calculated by multiplying the probability of at least one error by the total number of pages:

• \[ 1000 \times (1 - e^{-0.4}) \]

This step efficiently uses the principles of Poisson distribution to predict the number of faulty pages.

When dealing with large numbers, such as 1000 pages, the computation with Poisson distribution can become cumbersome. That's when normal approximation becomes useful. It provides a simpler way to estimate probabilities by substituting the Poisson model with a normal distribution model.
To apply normal approximation, we compute the mean and variance first.

• The mean, \( \lambda \), is already determined as:\[ \lambda = 1000 \times (1 - e^{-0.4}) \]
• The variance for a Poisson approximation will also be the same as the mean.

In this binomial approximation scenario, we calculate variance differently:

• \( \text{variance} = 1000 \times 0.4 \times (0.6) \)

To find the probability of more than 350 pages having errors, we convert this problem into a standard normal distribution problem using Z-scores. The Z-score is computed as:

• \[ Z = \frac{350 - \lambda}{\sqrt{240}} \]

This transformation allows us to use standard normal distribution tables to find the desired probability, making computations with large datasets manageable.

Calculating probability is a fundamental outcome in any statistical analysis. In our example, the task involves finding the probability that more than 350 out of 1000 pages have errors using normal approximation principles. By converting our Poisson model to a normal model, we use the Z-score to find the probability.
The Z-score conversion is given by the formula:

• \[ Z = \frac{350 - \lambda}{\sqrt{240}} \]

Where \( \lambda \) is our calculated mean for pages with errors. Here, the Z-score tells us how many standard deviations our 350-page estimate is from the mean of the distribution.
Once the Z-score is determined, it can be looked up in a standard normal distribution table or computed using statistical software to give us the probability of more than 350 pages having errors. This method of probability estimation leverages normal approximation, making it a powerful tool for handling large and complex datasets.