91Ó°ÊÓ

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

Understanding the Problem

We are given a normally distributed signal with a mean \( \mu = 1.5 \mathrm{~V} \) and a standard deviation \( \sigma = 0.02 \mathrm{~V} \). We need to calculate probabilities and value thresholds based on this distribution.

02

Probability within Specifications

We need to find the probability that a signal value is between \(1.45\mathrm{~V}\) and \(1.55\mathrm{~V}\). Transform these values to standard normal z-scores: \[ z = \frac{x - \mu}{\sigma} \] For \(x = 1.45\), \( z_1 = \frac{1.45 - 1.5}{0.02} = -2.5 \). For \(x = 1.55\), \( z_2 = \frac{1.55 - 1.5}{0.02} = 2.5 \). Look up these z-scores in the standard normal distribution table or use a calculator: \( P(z \leq 2.5) - P(z \leq -2.5) \approx 0.9876 - 0.0062 = 0.9814 \).

03

Calculating Signal Value Exceeding with 95% Probability

To find the voltage that is exceeded with \(95\%\) probability, find the 5th percentile of the standard normal distribution, which is \( z_{0.05} \approx -1.645 \). Then convert this back to a voltage: \[ x = \mu + z \cdot \sigma = 1.5 + (-1.645) \times 0.02 = 1.5 - 0.0329 = 1.4671 \mathrm{~V} \].

04

Probability of Exceeding Mean by Two Standard Deviations

We need the probability of a signal exceeding \(2\) standard deviations above the mean: \(x = \mu + 2\sigma = 1.5 + 2 \times 0.02 = 1.54 \mathrm{~V}\). Calculate the z-score: \( z = \frac{1.54 - 1.5}{0.02} = 2 \). Find \( P(Z > 2) \) from the standard normal table: \(1 - P(Z \leq 2) = 1 - 0.9772 = 0.0228\).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Normal Distribution

In probability and statistics, the normal distribution is a fundamental concept crucial for engineering applications. It is often referred to as the Gaussian distribution. This distribution is depicted graphically as a bell-shaped curve. The importance of the normal distribution lies in its ability to describe many natural phenomena and measurement data in engineering.

The curve is symmetric around the mean, implying that data is equally distributed on either side. Two parameters define a normal distribution: the mean ( mu ) and the standard deviation ( sigma ). The mean indicates the center of the distribution, while the standard deviation measures variability. Engineers utilize the normal distribution to predict outcomes, assess variability, and ensure quality control.

In electronic prognostics, understanding the normal distribution helps forecast system performance and detect potential faults.

Standard Deviation

Standard deviation is a statistical measure that quantifies the dispersion or spread of a set of data points. In the context of a normal distribution, it indicates how much values deviate from the mean, serving as a critical component in assessing the reliability of systems.

A smaller standard deviation indicates that data points are closer to the mean, suggesting consistent performance in engineering contexts. Conversely, a larger standard deviation signifies greater variability and potential unpredictability.

In the given exercise, a standard deviation of 0.02V signifies a very tight clustering of voltage values around the mean of 1.5V. This is important in precision engineering applications, where small deviations could lead to significant system impacts.

Z-score Calculation

The Z-score is a statistical measurement that describes a value's position relative to the mean of a group of values, expressed in terms of standard deviations. It helps in determining how unusual or typical a value is within a distribution.

To calculate the Z-score, use the formula:

• \( z = \frac{x - \mu}{\sigma} \)

Where \( x \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

A positive Z-score indicates the value is above the mean, while a negative Z-score indicates it is below. By standardizing values in a normal distribution, engineers can assess how far away a particular measurement is from what is typical or expected. This is pivotal for identifying anomalies in measurements during system prognostics.

Electronic Prognostics

Electronic prognostics is an advanced field that involves predicting the future reliability and potential failures of electronic systems. Using data collected from sensors and historical system telemetry, engineers apply statistical methods to detect when a system might fail.

By leveraging the normal distribution and standard deviation, prognostics can identify when a system operates outside its normal parameters. This proactive approach helps minimize downtime and reduce the requirement for unscheduled maintenance, ultimately saving time and resources.

Techniques like pattern recognition and early fault detection enable systems to learn their operational baselines and recognize deviations, ensuring high reliability and optimal performance in engineering. Prognostics combine predictive analytics and real-time data monitoring to enhance decision-making processes in critical systems.