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An article in Microelectronics Reliability ["Advanced Electronic Prognostics through System Telemetry and Pattern Recognition Methods" \((2007,\) Vol. \(47(12),\) pp. \(1865-1873)]\) presented an example of electronic prognosis. The objective was to detect faults to decrease the system downtime and the number of unplanned repairs in high-reliability systems. Previous measurements of the power supply indicated that the signal is normally distributed with a mean of \(1.5 \mathrm{~V}\) and a standard deviation of \(0.02 \mathrm{~V}\). a. Suppose that lower and upper limits of the predetermined specifications are \(1.45 \mathrm{~V}\) and \(1.55 \mathrm{~V},\) respectively. What is the probability that a signal is within these specifications? b. What is the signal value that is exceeded with \(95 \%\) probability? c. What is the probability that a signal value exceeds the mean by two or more standard deviations?

Step by step solution

01

Understand the Problem

We are given a normally distributed signal with a mean (\(\mu\)) of \(1.5\, \mathrm{V}\) and a standard deviation (\(\sigma\)) of \(0.02\, \mathrm{V}\). We need to solve three sub-questions involving probabilities and specific signal values under this normal distribution.

02

Calculate Probability within Specifications (Part a)

To find the probability that the signal is between \(1.45\, \mathrm{V}\) and \(1.55\, \mathrm{V}\), we need to compute the Z-scores for these values and use the standard normal distribution table.1. Compute Z-scores using the formula: \(Z = \frac{X - \mu}{\sigma}\). - For \(1.45\, \mathrm{V}\): \[Z_{1} = \frac{1.45 - 1.5}{0.02} = -2.5\] - For \(1.55\, \mathrm{V}\): \[Z_{2} = \frac{1.55 - 1.5}{0.02} = 2.5\]2. Use the standard normal distribution to find probabilities: - \(P(Z < 2.5) \approx 0.9938\) - \(P(Z < -2.5) \approx 0.0062\)3. Probability within specifications: \[P(1.45 < X < 1.55) = P(Z < 2.5) - P(Z < -2.5)\] \[= 0.9938 - 0.0062 = 0.9876\]

03

Determine Signal Value Exceeded with 95% Probability (Part b)

To find the signal value exceeded by 95% probability, we need the value corresponding to the 5th percentile (\(Z = -1.645\) approximately) of the standard normal distribution using the inverse Z formula:\[X = \mu + Z \cdot \sigma\]\[X = 1.5 + (-1.645) \times 0.02 = 1.467\, \mathrm{V}\]

04

Calculate Probability of Exceeding Mean by Two Standard Deviations (Part c)

Two standard deviations above the mean is calculated as:\[X = \mu + 2 \times \sigma = 1.5 + 2 \times 0.02 = 1.54\, \mathrm{V}\]To calculate this probability:1. Find the corresponding Z-score: \[Z = \frac{1.54 - 1.5}{0.02} = 2\]2. Use the standard normal distribution table to find: \(P(Z > 2) = 1 - P(Z < 2) \approx 1 - 0.9772 = 0.0228\)

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Key Concepts

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

Electronic Prognostics

Electronic prognostics refers to the process of predicting the future reliability and performance of electronic components and systems. This prediction enables proactive maintenance, which increases system uptime and decreases unscheduled repairs. In the microelectronics article mentioned, electronic prognostics aim to identify potential faults before they occur by analyzing existing data patterns in the system telemetry.

• **Purpose**: To enhance operational efficiency by reducing unexpected failures.
• **Methods**: Utilize pattern recognition and data analysis to forecast reliability.
• **Benefits**: Improved planning and reduced maintenance costs.

By analyzing data from the systems, such as power supply signals, and understanding the distribution of such data, engineers can predict when components might fail and take preventive actions.

Z-scores

Understanding Z-scores is crucial in normal distribution analysis. A Z-score measures how many standard deviations a data point is from the mean. It's a standard tool used to determine where a specific value stands in relation to the normal distribution.

To calculate a Z-score:

• Use the formula: \( Z = \frac{X - \mu}{\sigma} \)
• \(X\) is the value of the data point.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation.

In the context of electronic prognostics, Z-scores help determine the likelihood of system signals falling within certain specifications or limits, ideal for maintenance planning. In the example given, Z-scores are used to find the probability of signal values falling between the specified limits, assisting in fault detection.

Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. In the context of normal distribution, it describes how spread out the values are around the mean.

• **Significance**: Lower standard deviation means data points are close to the mean, while a higher one indicates more spread.
• **Formula**: Calculated as the square root of the variance.
• **Role in Normal Distribution**: The standard deviation defines the shape of the bell curve, with most values falling within one standard deviation of the mean.

In electronic prognostics, understanding the standard deviation of signals like voltage helps in defining acceptable ranges for operation, thus aligning maintenance strategies with performance expectations.

Probability Calculations

Probability calculations in the context of normal distribution help determine the likelihood of observing certain outcomes within specified limits. Using the Z-score, probabilities can be calculated by referring to standard normal distribution tables.

• **Z-score method**: Converts real-world problems into a standard form.
• **Distribution Table**: Used to find the probability associated with a Z-score.
• **Applications**: Helps in calculating expectations for electronic signal values to detect possible outliers indicating faults.

In the exercise provided, probability calculations are used to determine whether signals are within the given voltage specifications and to find values not typically exceeded, assisting in the early detection and prediction of potential system issues.