91Ó°ÊÓ

To understand how to calculate the error rate, consider the Poisson distribution, which is a statistical tool used to model the number of events that happen over a fixed interval of time or space. In the context of this exercise, the errors occurring along a magnetic recording surface are such events.

Given that the mean number of errors is stated as one error per \(10^5\) bits, we first need to convert the sector's data size to bits to find the expected number of errors in a single sector. Each sector contains 4096 eight-bit bytes, which can be calculated as:

• \(4096 \times 8 = 32768\) bits.

The mean number of errors expected in such a sector becomes the proportion of its size to the standard error rate:

• \(\lambda = \frac{32768}{10^5} = 0.32768\).

This \(\lambda\) represents the average number of errors we can anticipate in a given sector of 4096 bytes.

Once the mean number of errors (\(\lambda\)) is computed, the next step is to find the probability of certain numbers of errors occurring within the sector. This is done using the Poisson probability formula given by:

• \(P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\)

where \(k\) is the exact number of errors you want to find the probability for. Let's compute the probability of zero and one error occurring in the sector. This can be broken down as follows:

• The probability of zero errors (\(X = 0\)): \(P(X = 0) = e^{-0.32768} \approx 0.7206\).
• The probability of one error (\(X = 1\)): \(P(X = 1) = 0.32768 \times e^{-0.32768} \approx 0.2360\).

To find the probability of more than one error occurring (\(X > 1\)), we calculate:

• \(P(X > 1) = 1 - P(X = 0) - P(X = 1) = 1 - 0.7206 - 0.2360 = 0.0434\)

Thus, the probability of experiencing more than one error in a sector is approximately 4.34%.

The geometric distribution comes into play when determining how many sectors need to be evaluated until encountering the first error. A key insight into this is understanding it concerns independent trials, much like flipping a coin, where each flip is independent of previous flips.

We desire to know the mean number of sectors (trials) until at least one error presents itself, which aligns with calculating the mean of a geometric distribution since the condition here is the occurrence of at least one error per sector. The probability of having more than zero errors, which we denote as the success probability \(p\), is calculated as:

• \(P(X > 0) = 1 - P(X = 0) = 1 - 0.7206 = 0.2794\).

The mean number of trials until the first success is given by \(\frac{1}{p}\):

• Hence, \(\frac{1}{0.2794} \approx 3.58\)

This means, on average, you would expect approximately 3.58 sectors to pass before detecting the first error, providing a useful measure for error anticipation.