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What type of random variables can assume a countable infinity of values?
What type of random variables can assume a countable infinity of values?
Which property differentiates the cumulative distribution function (CDF) from other distribution functions?
Which property differentiates the cumulative distribution function (CDF) from other distribution functions?
What is the probability of not rolling a 1 with a fair six-sided dice?
What is the probability of not rolling a 1 with a fair six-sided dice?
In the context of normal distribution, what does the variable µ represent?
In the context of normal distribution, what does the variable µ represent?
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What does the complement rule state?
What does the complement rule state?
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What characteristic does a lognormal distribution exhibit when the standard deviation (σ) decreases?
What characteristic does a lognormal distribution exhibit when the standard deviation (σ) decreases?
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What is the probability that both light bulbs are defective according to the calculations provided?
What is the probability that both light bulbs are defective according to the calculations provided?
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What is the probability that a random customer purchases a brand 1’s TV and has to return it for repair under warranty?
What is the probability that a random customer purchases a brand 1’s TV and has to return it for repair under warranty?
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Which distribution is specifically suited for modeling lifetime variables governed by fatigue processes?
Which distribution is specifically suited for modeling lifetime variables governed by fatigue processes?
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Given events A and B that are mutually exclusive, how is their combined probability calculated?
Given events A and B that are mutually exclusive, how is their combined probability calculated?
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Which of the following statements about the properties of the normal distribution is true?
Which of the following statements about the properties of the normal distribution is true?
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Using the Law of Total Probability, what is the total probability that a TV sold will be returned for warranty repair work?
Using the Law of Total Probability, what is the total probability that a TV sold will be returned for warranty repair work?
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What is the correct statement about conditional probabilities P(B|A) and P(A|B)?
What is the correct statement about conditional probabilities P(B|A) and P(A|B)?
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What relationship does the lognormal distribution have with the normal distribution?
What relationship does the lognormal distribution have with the normal distribution?
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What is the outcome of this conditional probability scenario: A = {the number is greater than 3} and B = {the number is even}?
What is the outcome of this conditional probability scenario: A = {the number is greater than 3} and B = {the number is even}?
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Which of the following is a characteristic of discrete probability distribution?
Which of the following is a characteristic of discrete probability distribution?
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If two events A and B are independent, what can be inferred about their probabilities?
If two events A and B are independent, what can be inferred about their probabilities?
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Which of the following best describes the hazard rate function in the context of reliability engineering?
Which of the following best describes the hazard rate function in the context of reliability engineering?
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In the context of warranty repair work, what probability represents the likelihood of a brand 2 TV being returned?
In the context of warranty repair work, what probability represents the likelihood of a brand 2 TV being returned?
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In an example where 5 defective light bulbs are packed with 20 good ones, what is the probability of randomly selecting one defective bulb?
In an example where 5 defective light bulbs are packed with 20 good ones, what is the probability of randomly selecting one defective bulb?
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What does the notation P(B|A) signify in probability theory?
What does the notation P(B|A) signify in probability theory?
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How is the probability of getting an even number when rolling a die that shows a number greater than 3 calculated?
How is the probability of getting an even number when rolling a die that shows a number greater than 3 calculated?
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What is the probability of failure, F(t), referring to?
What is the probability of failure, F(t), referring to?
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Which of the following probabilities represents the chance of returning a brand 3 TV for repair?
Which of the following probabilities represents the chance of returning a brand 3 TV for repair?
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What is the primary purpose of the Laplace transform in engineering applications?
What is the primary purpose of the Laplace transform in engineering applications?
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How is the reliability of equipment over a specified time calculated when the time to failure is exponentially distributed?
How is the reliability of equipment over a specified time calculated when the time to failure is exponentially distributed?
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Which mathematical property defines the cumulative distribution function (CDF) for a continuous random variable?
Which mathematical property defines the cumulative distribution function (CDF) for a continuous random variable?
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What is the variance of a continuous random variable X defined by a probability density function?
What is the variance of a continuous random variable X defined by a probability density function?
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Given a failure rate of 0.003 failures per hour, what is the reliability of the equipment during a 10-hour mission calculated using exponential distribution?
Given a failure rate of 0.003 failures per hour, what is the reliability of the equipment during a 10-hour mission calculated using exponential distribution?
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What does the probability density function (PDF) describe?
What does the probability density function (PDF) describe?
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In reliability engineering, what aspect does the exponential distribution model primarily address?
In reliability engineering, what aspect does the exponential distribution model primarily address?
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What does the term 'failure rate' specifically refer to in the context of exponential distribution?
What does the term 'failure rate' specifically refer to in the context of exponential distribution?
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Study Notes
CGE676 Reliability and Maintenance Engineering - Chapter 2: Statistical Reliability
- This chapter covers statistical reliability, specifically probability and its applications in reliability.
Rules of Probability
- Probability of an event happening = (Number of ways it can happen) / (Total number of outcomes)
- Examples provided include coin tosses and rolling dice to illustrate the concept.
- The probability of mutually exclusive events is zero (cannot happen together).
- The probability of A or B equals the probability of A plus the probability of B
Basic Statistics in Reliability - Probability
- Conditional Probability: P(A|B) = P(A and B) / P(B).
- Conditional probability is the probability of an event (A) happening given another event (B) has already happened.
- The probability of event A given event B has happened is different from the probability of event B given event A has happened (P(A|B) ≠ P(B|A)).
- A and B are independent if P(A and B) = P(A) * P(B)
- Example: Probability of getting an even number on a dice given it is greater than 3
Multiplicative Rule
- P(A and B) = P(A|B) * P(B) = P(B|A) * P(A)
- Events A and B are independent if and only if P(A and B) = P(A) * P(B)
- Example: Probability of selecting two defective lightbulbs from a box of defective and non-defective light bulbs
- Important Note: The selection is without replacement.
Law of Total Probability (LTP)
- If A1, A2, ..., An are mutually exclusive and exhaustive events in sample space S, then for any other event B in S: P(B) = Σ P(B|Aᵢ)P(Aᵢ)
- Example: Probability of a random TV requiring warranty repair, given the three brands of TVs each with different market shares.
Probability of Failure, F(t)
- F(t)=1-R(t)=1-∫ exp(-λ(x))dx
Random Variables
- A variable whose value changes.
- Random variables are a set of possible values from a random experiment.
- Example: Tossing a coin - possible values are Heads or Tails
Probability Distribution
- Classified into discrete and continuous
- Discrete Variables: Involve counting (e.g., number of students, red marbles, coin tosses)
- Example: Toss a coin twice, count tails.
- Continuous Variables: Involve measuring (e.g., height, weight, time, distance)
- Example: Height of students, weight of students in a class.
Cumulative Distribution Function (CDF)
-
Calculates the cumulative probability of a given value X (P(X ≤ x)).
-
Example: Toss two dice, summing the result. Finding the cumulative probability of summing a certain number.
Probability Density Function (PDF)
-
Defines the probability function representing the density of a continuous random variable.
-
Properties:
- 0 ≤ f(x) ≤ 1
- ∫₋∞⁺∞ f(x)dx = 1
- P(a ≤ X ≤ b) = ∫ₐᵇ f(x)dx
-
Example 3: Probability density function for a random variable X, where f (x)= e-x is given if X ≥ 0, zero otherwise.
Cumulative Distribution Function (CDF)
- Properties:
- f(x) = d F(x) / dx,
- lim F(x) = 0 and lim F(x) = 1 as x → -∞ and x → ∞
- P(x₁ ≤ x ≤ x₂) = F(x₂) - F(x₁) if x₁ < x₂
Relationship between Measures
- Provides relationships between Probability of Failure, Reliability, Probability Density Function, and Failure Rate. A table shows the formulas for these different measures
Exponential Distribution
- A widely used probability distribution.
-
Properties:
- Probability density function: f(t) = λe(-λt) for t ≥ 0, λ > 0.
- Probability of failure: F(t) = 1 - e(-λt)
- Reliability: R(t) = e(-λt)
Applications & Examples
- Various examples of applications of these concepts in reliability engineering (e.g., equipment failure within a certain time frame)
Laplace Transform
- A useful method for solving differential equations, especially when initial values are zero.
- Formulas for various functions (e.g., constants, e-at, etc.) are given in tabular format.
Next Lecture
- The next lecture will be focused on Reliability Networks
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Description
Test your understanding of statistical reliability concepts covered in Chapter 2 of CGE676. This quiz focuses on the rules of probability, including conditional probabilities and examples involving coin tosses and rolling dice. It's perfect for reinforcing your grasp of how probability applies to reliability engineering.
