Problem 25 Beim Korrekturlesen eines Manusk... [FREE SOLUTION]

91影视

Elementare Wahrscheinlichkeitstheorie und stochastische Prozesse

Kai Lai Chung

$Math Studyset 91影视 Explanations$ Math

1978 Edition

Chapter 8: Problem 25

Beim Korrekturlesen eines Manuskripts findet jeder Leser mindestens einen Fehler. Wenn jedoch zu Beginn j Fehler vorhanden sind, dann l盲Bt er eine Anzahl von Fehlern darin, die von 0 bis $j-1$ gleichverteilt ist. Wie gro脽 ist der Erwartungswert fur die Anzahl von Lesern, die man braucht, um alle Fehler zu entdecken? (Hinweis: $e_{j}=\frac{1}{j}\left(e_{1}+\ldots+e_{j-1}\right)+1 ;$ nun vereinfache man $\left.e_{j}-e_{j-1} \cdot\right)$

Short Answer

Expert verified

The expected number of readers is the j-th harmonic number, $ H_j $.

Step by step solution

Understanding the Problem

Each reader finds at least one error. Initially, there are j errors. The number of errors found by a reader is uniformly distributed between 0 and j-1. We need to find the expected number of readers required to find all errors.

Using the Given Hint

The hint provides the equation: \[ e_j = \frac{1}{j} (e_1 + e_2 + ... + e_{j-1}) + 1 \]which will be used to simplify and find the expectation value.

Consider Simplifying

To simplify, subtract the equations for $ e_j $ and $ e_{j-1} $: \[ e_j - e_{j-1} = \frac{1}{j} (e_1 + e_2 + ... + e_{j-1}) + 1 - e_{j-1} \]

Simplifying Further

Notice that $(e_1 + e_2 + ... + e_{j-2}) $ cancels out so we get: \[ e_j - e_{j-1} = \frac{1}{j} e_{j-1} + \frac{1}{j} (j-1) \]

Final Simplification

Reorganize the equation: \[ e_j - e_{j-1} = \frac{e_{j-1}}{j} + \frac{j-1}{j} \] \[ e_j - e_{j-1} = \frac{e_{j-1} + (j-1)}{j} \]

Summing Differences

Summing the difference from $ e_2 $ to $ e_j $: \[ e_j - e_1 = \sum_{k=2}^j \frac{e_{k-1} + (k-1)}{k} \]

Relating to Harmonic Numbers

Recognize that summing the harmonic series and rearranging terms, leading to: \[ e_j = H_j \] where $ H_j = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{j} $

Conclusion

Thus, the expectation value for the number of readers needed is equivalent to the j-th harmonic number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Uniform Distribution

In the problem, the errors found by each reader are uniformly distributed between 0 and $j-1$. This means each integer in this range has an equal chance of being selected. For example:

If there are initially 3 errors, then the number of errors a reader might find could be 0, 1, or 2 with equal probability.

The uniform distribution is a common concept in probability theory, where every outcome in a given range is equally likely. This concept makes it easier to compute expected values, probabilities, and variances.

Harmonic Series

The harmonic series is a sequence of numbers formed by the sum of the reciprocals of the positive integers. It is given by:
\[ H_n = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} \]
In this problem, the expected number of readers required to find all errors, denoted by $e_j$, corresponds to the j-th harmonic number.
Harmonic numbers grow logarithmically, which means they increase very slowly even as j becomes large. Therefore, to resolve all errors, you can expect to have to check a number of readers corresponding to $H_j$.
This is important as it gives us a practical idea of the effort required.

Expected Number of Trials

The expected value is a central concept in probability theory and describes the average outcome if the experiment is repeated many times. For this problem,
we need to find the expected number of readers required to uncover all errors. Using the harmonic series, we establish that the expected number of trials (readers, in this case) to find all errors is given by the j-th harmonic number $H_j$.
This result provides the average number of readers you would expect to need in order to discover all errors in the manuscript.

Probability Theory

Probability theory is a branch of mathematics concerned with analyzing random phenomena. The foundation of probability theory includes:

Uniform distributions
Expected values
Probability distributions

In this exercise, we deal with the probability of finding errors uniformly distributed, where every error count between 0 to $j-1$ is equally likely for a reader to find. Probability theory allows us to calculate how long, on average, it will take (i.e., the expected number of readers) to find all the errors.
This is achievable by using notions like the harmonic series, which arise from summing the expected results of repeated independent trials.

Error Correction Process

The problem of error detection and correction is integral to fields like data transmission and coding theory. Errors must be identified and corrected to ensure the integrity of the transmitted or copied information.
In the manuscript checking problem, readers sequentially and independently look for errors, correcting them as they find them. The uniform distribution model suggests each reader might find any number of errors from 0 to $j-1$.
This error correction process is analogous to how it might take several rounds of proofreading by different individuals to catch all mistakes in a document, emphasizing the importance of repeated reviews in error detection systems.

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