Problem 1 - An Introduction to Mathematical Statistics and Its Applications

Question

If a typist averages one misspelling in every 3250 words, what are the chances that a 6000 -word report is free of all such errors? Answer the question two waysfirst, by using an exact binomial analysis, and second, by using a Poisson approximation. Does the similarity (or dissimilarity) of the two answers surprise you? Explain.

Step-by-Step Solution

Answer

The probability of a 6000-word report having no errors calculated using the exact binomial analysis is \((1-\frac{1}{3250})^{6000}\) and using the Poisson approximation it is \(e^{-(\frac{6000}{3250})}\). The two results are similar due to Poisson distribution approximation for large \(n\) and small \(p\).

Step 1: Calculate Exact Binomial Probability

The average error rate is \(p = \frac{1}{3250}\). In a 6000-word report, the number of 'trials' is 6000. We want to compute the probability that no errors, or 'successes', occur. The binomial probability is given by \(\displaystyle P(k;n,p) = \binom{n}{k}p^k(1-p)^{n-k}\), where \(k\) is the number of successes, \(n\) the number of trials, and \(p\) the error rate. Substituting the known values into the equation yields \(P(0;6000,\frac{1}{3250}) = \binom{6000}{0} (\frac{1}{3250})^0 (1-\frac{1}{3250})^{6000}\)

Step 2: Simplify Binomial Probability

Simplify the terms to get the probability as \((1- \frac{1}{3250})^{6000}\)

Step 3: Calculate the Poisson Approximation

For the Poisson approximation, the rate parameter \(\lambda\) is equal to \(np\) . For our problem, \(\lambda = 6000 * \frac{1}{3250}\). The formula for the Poisson probability is \(\displaystyle P(k;\lambda) = \frac{\lambda^k e^{-\lambda}}{k!}\). Here, \(k = 0\), so finding the Poisson probability leads to \(P(0; \frac{6000}{3250}) = \frac{(\frac{6000}{3250})^0 e^{-(\frac{6000}{3250})}}{0!}\).

Step 4: Simplify Poisson Approximation

Simplify the terms to get the Poisson approximation as \(e^{-(\frac{6000}{3250})}\).

Step 5: Evaluate the Similarity Between the Two Results

The two answers must appear quite similar because the Poisson distribution can approximate the binomial distribution well for large \(n\) and small \(p\).

Key Concepts

Binomial Distribution

The Binomial Distribution is a fundamental concept in probability theory used to model the number of successes in a specific number of independent trials, where each trial has a binary outcome: success or failure. Each trial has the same probability of success, denoted as \( p \). In our original exercise, the typist's average error rate is represented as the probability of making a misspelling, which is \( p = \frac{1}{3250} \). The number of 'trials' is the total number of words in the report, which is 6000.

To find out the probability of observing no errors in the 6000-word report, we use the binomial probability formula:
\[ P(k;n,p) = \binom{n}{k}p^k(1-p)^{n-k} \]
Here, \( k = 0 \), which means we are interested in the event where zero misspellings occur. The expression simplifies because \( p^0 = 1 \), and the binomial coefficient \( \binom{6000}{0} = 1 \). Thus, the expression reduces to \((1-p)^{6000}\).

The binomial distribution is useful for analyzing situations like this where specific event outcomes are predictable and each outcome is independent.

Poisson Distribution

The Poisson Distribution is especially useful in situations where we are interested in the number of events that occur in a fixed interval of time or space, under conditions where these events happen with a known constant mean rate and independently of the time since the last event. It's a good approximation for the binomial distribution, particularly when the number of trials is large and the probability of success is small.

In the context of our problem, we approximate the binomial distribution using the Poisson model, by calculating the rate parameter \( \lambda \), which equals \( np \) (here, \( \lambda = 6000 \times \frac{1}{3250} \)). The Poisson probability formula for zero errors is:
\[ P(k;\lambda) = \frac{\lambda^k e^{-\lambda}}{k!} \]
Since \( k = 0 \), the probability formula simplifies to \( e^{-\lambda} \).

The Poisson approximation is applicable and practical here, as it simplifies calculations that involve large datasets with rare events.

Statistical Approximation

Statistical Approximation is a method used to simplify complex mathematical calculations by substituting one form of distribution for another while maintaining a reasonable degree of accuracy. In our exercise, by employing Poisson approximation, we attempted to find the probability of no misspellings without directly using the binomial distribution.

Factors that make this approximation valid include:

• A high number of trials (6000 words in our example).
• A small probability of success (the low typist's error rate).

This methodology allows us to replace a computation-heavy analysis with a more manageable approach.

The approximation maintains accuracy due to the overlapping nature and conditions of distribution overlap, especially when \( n \) is large and \( p \) is small.

Error Rates

Error Rates are quantifications of the likelihood of incorrect outcomes, expressed as a fraction or proportion. In the context of our typist, the error rate is \( \frac{1}{3250} \), signifying that, on average, a misspelling occurs once in every 3250 words typed.

Understanding and analyzing error rates can be significant in quality control and ensuring high standards in processes such as content writing, data entry, or technical drafting.

• Helps in predicting the level of quality you may expect.
• Essential in assessing processes and finding areas of improvement.

The calculated probabilities (from Binomial and Poisson distributions) indicate how likely the document is clear of errors based on the known error rate.

Statistical Analysis

Statistical Analysis is the science of collecting, reviewing, and presenting data to identify patterns, relationships, and trends. It assists in the effective interpretation of data in probability and decision-making processes.

In our exercise, statistical analysis helps identify which model (binomial versus Poisson) best simplifies the problem without losing significative accuracy. This form of analysis is essential:

• To assess risk and reliability within datasets.
• To ensure that mathematical assumptions and approximations are valid.
• To apply the correct statistical methods according to the nature of the data and the desired outcomes.

It encompasses choosing between distributions for approximation, validating the error rates, and justifying results, ensuring practically sound conclusions and predictions are made.