Problem 83 - Applied Statistics and Probability for Engineers

Question

The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours. (a) What is the probability that a laser fails before 5000 hours? (b) What is the life in hours that \(95 \%\) of the lasers exceed? (c) If three lasers are used in a product and they are assumed to fail independently, what is the probability that all three are still operating after 7000 hours?

Step-by-Step Solution

Answer

(a) 0.0004; (b) 7987 hours; (c) 0.125.

Step 1: Understanding the Problem

We need to calculate the probabilities and time related to the lifespan of semiconductor lasers based on a normal distribution with a mean (\( \mu \)) of 7000 hours and a standard deviation (\( \sigma \)) of 600 hours. We'll solve for three different scenarios.

Step 2: Calculating Probability for Part (a)

To find the probability that a laser fails before 5000 hours, we need to calculate the Z-score for 5000 hours using the formula: \[ Z = \frac{X - \mu}{\sigma} \] where \( X = 5000 \), \( \mu = 7000 \), and \( \sigma = 600 \).Substitute the values:\[ Z = \frac{5000 - 7000}{600} = \frac{-2000}{600} = -3.33 \]Using the Z-table, find the probability for \( Z = -3.33 \). This corresponds to a probability of approximately 0.0004.

Step 3: Calculating Life Hours for Part (b)

To find the life in hours that 95% of the lasers exceed, we need to find the Z-score corresponding to the upper 5% of the normal distribution, which is approximately \( Z = 1.645 \).We use the inverse Z-score formula:\[ X = \mu + Z \cdot \sigma \]Substitute the values:\[ X = 7000 + 1.645 \cdot 600 = 7000 + 987 = 7987 \]Therefore, 95% of the lasers exceed 7987 hours.

Step 4: Calculating Probability for Part (c)

To find the probability that all three lasers are still operating after 7000 hours, first calculate the probability of a single laser operating beyond 7000 hours, which corresponds to \( Z = 0 \). The probability from the Z-table is 0.5, meaning there is a 50% chance a single laser will exceed 7000 hours.Since the lasers operate independently, the probability for all three exceeding 7000 hours is given by:\( (0.5)^3 = 0.125 \).

Step 5: Conclusion

Combine the results: (a) The probability that a laser fails before 5000 hours is 0.0004. (b) 95% of the lasers exceed 7987 hours. (c) The probability that all three lasers operate past 7000 hours is 0.125.

Key Concepts

Semiconductor Laser

Semiconductor lasers are a specific type of laser that uses a semiconductor as its gain medium. Lasers are devices that emit light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. Semiconductor lasers are commonly used because they can be made in very small sizes and are very efficient. Their typical applications include telecommunications, barcode readers, laser pointers, and in the manufacture of consumer electronics. These lasers can vary widely in lifespan, depending on their quality and usage conditions. Understanding their longevity is crucial for both manufacturers and users.
This is because it affects factors like the frequency of replacing the devices or the overall durability of an entire product that uses such lasers. The lifespan of semiconductor lasers often follows a normal distribution, which means their lifetimes tend to cluster around a mean value with a predictable variability. This nature allows for statistical lifespan analysis to predict and enhance performance and maintenance schedules.

Z-score

A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean. The formula used to calculate a Z-score is: \[ Z = \frac{X - \mu}{\sigma} \]where:

• \(X\) is the value in question.
• \(\mu\) is the mean of the data set.
• \(\sigma\) is the standard deviation of the data set.

In our exercise, Z-scores help determine the probability of the semiconductor laser failing before a certain number of hours. Negative Z-scores represent values below the mean, while positive Z-scores indicate values above the mean. Calculating these scores can help identify how rare or common a certain lifespan is, helping make informed decisions about product reliability.

Probability Calculation

Probability calculation involves determining the likelihood of a given event to happen. In statistics, especially when dealing with normally distributed data, probabilities can be computed using the Z-score tables (also called standard normal distribution tables). Once you have a Z-score, you can look it up in these tables to find the corresponding probability. In task (a) of the provided exercise, we are finding the probability that a laser fails before 5000 hours. This was achieved by finding a Z-score of -3.33.
Using the Z-score table, a probability of 0.0004 was calculated, implying such a failure is very rare. In contrast, in task (c), the probability that all three lasers will last beyond the mean life was calculated using the powers of individual likelihoods due to the independence of laser function. Understanding these probabilities allows businesses to assess risks and design products that meet durability expectations.

Lifespan Analysis

Lifespan analysis is the study of how long a device or component can be expected to function before it fails. For semiconductor lasers, this is a crucial aspect because reliability is often a key requirement for their applications. The concept of lifespan in this context is related to mean lifespan and variability, represented by the standard deviation, within a normally distributed population. The given exercise provided a normal distribution of a mean lifespan of 7000 hours and a standard deviation of 600 hours. In part (b) of the exercise, lifespan analysis involved finding that 95% of the lasers live beyond 7987 hours. This was determined by identifying a Z-score for the desired percentile (1.645) and using it to calculate the hours exceeded by 95% of the lasers. Analyzing lifespans based on such distributions helps determine the expected time frame for functional operation before any failure occurs, which can directly influence product maintenance schedules and warranty policies.