CGE676 Chapter 2
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Questions and Answers

What type of random variables can assume a countable infinity of values?

  • Unimodal random variables
  • Continuous random variables
  • Discrete random variables (correct)
  • Lognormal random variables

Which property differentiates the cumulative distribution function (CDF) from other distribution functions?

  • Area under the curve equals 1. (correct)
  • It is asymptotic to the vertical axis.
  • It only applies to discrete distributions.
  • It can take negative values.

What is the probability of not rolling a 1 with a fair six-sided dice?

  • 2/3
  • 1/6
  • 1/2
  • 5/6 (correct)

In the context of normal distribution, what does the variable µ represent?

<p>Mean value (D)</p> Signup and view all the answers

What does the complement rule state?

<p>P(A') = 1 - P(A) (D)</p> Signup and view all the answers

What characteristic does a lognormal distribution exhibit when the standard deviation (σ) decreases?

<p>It approaches the characteristics of a normal distribution. (B)</p> Signup and view all the answers

What is the probability that both light bulbs are defective according to the calculations provided?

<p>1/30 (B)</p> Signup and view all the answers

What is the probability that a random customer purchases a brand 1’s TV and has to return it for repair under warranty?

<p>0.0125 (A)</p> Signup and view all the answers

Which distribution is specifically suited for modeling lifetime variables governed by fatigue processes?

<p>Lognormal distribution (C)</p> Signup and view all the answers

Given events A and B that are mutually exclusive, how is their combined probability calculated?

<p>P(A U B) = P(A) + P(B) (C)</p> Signup and view all the answers

Which of the following statements about the properties of the normal distribution is true?

<p>It has a maximum at the mean µ. (B)</p> Signup and view all the answers

Using the Law of Total Probability, what is the total probability that a TV sold will be returned for warranty repair work?

<p>0.0205 (A)</p> Signup and view all the answers

What is the correct statement about conditional probabilities P(B|A) and P(A|B)?

<p>P(B|A) ≠ P(A|B) (D)</p> Signup and view all the answers

What relationship does the lognormal distribution have with the normal distribution?

<p>It is derived from the normal distribution through logarithmic transformation. (A)</p> Signup and view all the answers

What is the outcome of this conditional probability scenario: A = {the number is greater than 3} and B = {the number is even}?

<p>P(B|A) = 2/3 (D)</p> Signup and view all the answers

Which of the following is a characteristic of discrete probability distribution?

<p>Involves counting specific events (C)</p> Signup and view all the answers

If two events A and B are independent, what can be inferred about their probabilities?

<p>P(A|B) = P(A) (D)</p> Signup and view all the answers

Which of the following best describes the hazard rate function in the context of reliability engineering?

<p>It describes the failure rate over time. (D)</p> Signup and view all the answers

In the context of warranty repair work, what probability represents the likelihood of a brand 2 TV being returned?

<p>0.006 (A), 0.0060 (C)</p> Signup and view all the answers

In an example where 5 defective light bulbs are packed with 20 good ones, what is the probability of randomly selecting one defective bulb?

<p>1/5 (B), 1/5 (D)</p> Signup and view all the answers

What does the notation P(B|A) signify in probability theory?

<p>The conditional probability of event B given event A (B)</p> Signup and view all the answers

How is the probability of getting an even number when rolling a die that shows a number greater than 3 calculated?

<p>2/3 (C)</p> Signup and view all the answers

What is the probability of failure, F(t), referring to?

<p>It indicates random variable behavior. (C)</p> Signup and view all the answers

Which of the following probabilities represents the chance of returning a brand 3 TV for repair?

<p>0.0020 (A), 0.002 (B)</p> Signup and view all the answers

What is the primary purpose of the Laplace transform in engineering applications?

<p>To provide a method for solving differential equations (A)</p> Signup and view all the answers

How is the reliability of equipment over a specified time calculated when the time to failure is exponentially distributed?

<p>By computing the failure rate over that duration (C)</p> Signup and view all the answers

Which mathematical property defines the cumulative distribution function (CDF) for a continuous random variable?

<p>It describes the probability that a random variable is less than or equal to a certain value (A), It can never decrease as x increases (B)</p> Signup and view all the answers

What is the variance of a continuous random variable X defined by a probability density function?

<p>It is given by var(X) or σX² (B)</p> Signup and view all the answers

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?

<p>Approximately 97% (D)</p> Signup and view all the answers

What does the probability density function (PDF) describe?

<p>The likelihood of a random variable taking on specific values (A)</p> Signup and view all the answers

In reliability engineering, what aspect does the exponential distribution model primarily address?

<p>The time until the first failure of a system (A)</p> Signup and view all the answers

What does the term 'failure rate' specifically refer to in the context of exponential distribution?

<p>The inverse of the average time between failures (A)</p> Signup and view all the answers

Flashcards

Probability of an event

The likelihood or chance of an event occurring, expressed as a number between 0 and 1.

Mutually Exclusive Events

Events that cannot occur at the same time.

Conditional Probability

The probability of an event occurring given that another event has already occurred.

Independent Events

Events whose occurrence does not affect each other's probability.

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Complement Rule

The probability of an event not occurring is 1 minus the probability of it occurring.

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Multiplicative Rule

The probability of two events occurring in sequence is the product of their individual probabilities (if independent).

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Sample Space

The set of all possible outcomes in a random experiment.

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Probability of an event (P(A))

A numerical value between 0 and 1 (inclusive) representing how likely an event A is to occur. 0 means impossible; 1 means it is certain to occur.

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Probability of Defective Light Bulbs

The likelihood of both removed light bulbs being faulty in a sequence.

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Law of Total Probability

Used to find the overall probability of an event resulting from multiple, distinct situations.

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Random Variable

A variable whose value is a numerical outcome of a random phenomenon.

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Discrete Probability Distribution

Probability distribution used to describe possible outcomes of a discrete random variable.

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Continuous Probability Distribution

Probability distribution used to describe possible outcomes of a continuous random variable.

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Probability of Failure (F(t))

The likelihood or probability that a particular item will fail at a given time.

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Discrete Random Variable

A variable whose value can only be whole numbers, often obtained by counting. For example, the number of heads when flipping a coin.

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Continuous Random Variable

A variable whose value can take on any number within a range, often obtained by measuring. For example, the height of a person.

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Cumulative Distribution Function (CDF)

A function that describes the probability of a random variable being less than or equal to a certain value. It ranges from 0 to 1 and represents the accumulated probability up to a specific point.

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Normal Distribution

A bell-shaped probability distribution that is symmetrical around the mean. Many natural phenomena follow this pattern.

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Normal Distribution in Reliability

Describes the lifetime behavior of components that experience wear-out over time. The hazard rate function helps identify the rate of failure as a function of time.

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Lognormal Distribution

A distribution where the natural logarithm of the variable follows a normal distribution. It's skewed to the right and commonly used for lifetime data influenced by fatigue processes.

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Lognormal Distribution in Reliability

Fits lifetime data of components subjected to fatigue processes, like wear-out mechanisms. The hazard rate function reflects the probability of failure over time.

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Properties of Lognormal Distribution

A positive-valued, skewed distribution with a strong connection to the normal distribution. Its shape becomes more symmetric as the standard deviation decreases.

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Exponential distribution

A probability distribution commonly used in reliability and maintenance, describing the probability of an event occurring at any given time.

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Probability Density Function (PDF)

A function that describes the probability of a continuous random variable taking on a specific value. It's represented by f(x).

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Properties of PDF

The total area under the PDF curve is always 1, indicating that the probability of all possible values occurring is 100%.

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Cumulative Density Function (CDF)

A function representing the probability that a random variable takes on a value less than or equal to a given value. It's denoted as F(x).

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Properties of CDF

The CDF always increases from 0 to 1 as the value of x increases. It reflects the accumulated probability up to a given value.

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Reliability

The probability that a system or component will perform its intended function for a given period of time without failing.

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Laplace Transform

A mathematical tool used to transform differential equations into simpler algebraic equations, particularly helpful when initial conditions are zero.

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Application in Reliability

The exponential distribution is frequently used to model the time until failure of equipment, allowing us to calculate the reliability of a system for a specific mission duration.

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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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CGE676 Chapter 2: Statistical Reliability PDF

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.

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