3.4 Common probability models in engineering

Probability models in engineering help us understand and predict random events. This section covers key distributions like Bernoulli, binomial, Poisson, uniform, exponential, and normal. Each model has unique characteristics suited for different scenarios.

We'll learn how to choose the right model, estimate parameters, and apply these models to real engineering problems. This knowledge is crucial for analyzing data, making predictions, and solving complex engineering challenges involving uncertainty.

Discrete Probability Distributions in Engineering

Bernoulli and Binomial Distributions

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• The Bernoulli distribution is a discrete probability distribution for a random variable that can take on only two possible outcomes, typically labeled as success or failure, with a fixed probability of success (p) for each trial
• The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, where the probability of success remains constant across all trials
  • The probability mass function (PMF) of a binomial random variable X with parameters n and p is given by $P(X = k) = C(n, k) * p^k * (1-p)^(n-k)$, where $C(n, k)$ represents the number of ways to choose k items from a set of n items
  • Example: The number of defective items in a sample of 10 products from a manufacturing process with a 5% defect rate follows a binomial distribution with n = 10 and p = 0.05
  • Example: The number of successful launches in 5 attempts, where each launch has a 90% success rate, follows a binomial distribution with n = 5 and p = 0.9

Poisson Distribution

• The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring within a fixed interval of time or space, assuming these events occur with a known constant mean rate and independently of the time since the last event
  • The Poisson distribution is often used to model the number of occurrences of rare events, such as the number of defects in a manufacturing process or the number of customers arriving at a service desk within a specific time frame
  • The probability mass function (PMF) of a Poisson random variable X with parameter λ is given by $P(X = k) = (λ^k * e^(-λ)) / k!$, where λ represents the average number of events per interval and e is the base of the natural logarithm
  • Example: The number of traffic accidents at a particular intersection per month follows a Poisson distribution with an average of 2.5 accidents per month
  • Example: The number of customer complaints received by a call center per hour follows a Poisson distribution with an average of 4 complaints per hour

Continuous Probability Distributions

Uniform and Exponential Distributions

• The uniform distribution is a continuous probability distribution that describes a situation where all values within a given range [a, b] are equally likely to occur
  • The probability density function (PDF) of a uniform random variable X on the interval [a, b] is given by $f(x) = 1 / (b - a)$ for $a ≤ x ≤ b$, and $f(x) = 0$ otherwise
  • Example: The time a traffic light stays green follows a uniform distribution between 30 and 60 seconds
  • Example: The position of a randomly dropped pin on a 10-meter line segment follows a uniform distribution between 0 and 10 meters
• The exponential distribution is a continuous probability distribution that models the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate
  • The probability density function (PDF) of an exponential random variable X with parameter λ is given by $f(x) = λ * e^(-λx)$ for $x ≥ 0$, and $f(x) = 0$ for $x < 0$, where λ represents the average number of events per unit time
  • Example: The time between consecutive customer arrivals at a service desk follows an exponential distribution with an average of 10 minutes between arrivals
  • Example: The lifetime of a light bulb follows an exponential distribution with an average lifetime of 2000 hours

Normal Distribution

• The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about the mean and follows a bell-shaped curve
  • The normal distribution is characterized by two parameters: the mean (μ) and the standard deviation (σ), which determine the location and spread of the distribution, respectively
  • The probability density function (PDF) of a normal random variable X with parameters μ and σ is given by $f(x) = (1 / (σ * √(2π))) * e^(-(x - μ)^2 / (2σ^2))$, where π is the mathematical constant pi (approximately 3.14159)
  • The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1, often denoted as $Z ~ N(0, 1)$
  • Example: The heights of adult males in a population follow a normal distribution with a mean of 70 inches and a standard deviation of 3 inches
  • Example: The measurement errors of a precision instrument follow a normal distribution with a mean of 0 and a standard deviation of 0.01 units

Selecting Probability Models for Engineering

Identifying Key Characteristics and Assumptions

• Identify the key characteristics of the random variable or process under consideration, such as the type of variable (discrete or continuous), the range of possible values, and any underlying assumptions or constraints
• Determine whether the random variable or process exhibits memoryless property, which is a characteristic of the exponential distribution and indicates that the future state of the system depends only on the current state, not on the past
• Consider the physical or practical interpretation of the random variable or process to guide the selection of an appropriate probability model
  • For example, the Poisson distribution may be suitable for modeling the number of defects in a manufacturing process, while the exponential distribution may be appropriate for modeling the time between equipment failures

Assessing Data and Evaluating Goodness-of-Fit

• Assess the availability and quality of data related to the random variable or process, as this can influence the choice of probability model and the estimation of its parameters
• Evaluate the goodness-of-fit of the selected probability model using techniques such as probability plots, hypothesis tests, or model selection criteria (e.g., Akaike information criterion or Bayesian information criterion) to ensure that the model adequately describes the observed data
  • Example: Use a chi-square goodness-of-fit test to determine whether the observed frequencies of defects in a manufacturing process are consistent with a Poisson distribution
  • Example: Construct a normal probability plot to visually assess whether a dataset follows a normal distribution

Analyzing Engineering Problems with Probability

Formulating Problems and Estimating Parameters

• Formulate the engineering problem in terms of random variables and probability distributions, clearly defining the objectives, constraints, and performance metrics
• Identify the relevant probability models that best describe the random variables or processes involved in the problem, based on the available data, domain knowledge, and the problem's characteristics
• Estimate the parameters of the selected probability models using techniques such as maximum likelihood estimation, method of moments, or Bayesian inference, depending on the available data and prior knowledge
  • Example: Use maximum likelihood estimation to determine the parameter λ of a Poisson distribution based on the observed number of defects in a manufacturing process
  • Example: Apply the method of moments to estimate the mean and standard deviation of a normal distribution from a sample of measurements

Conducting Probabilistic Calculations and Interpreting Results

• Perform probabilistic calculations and derive key properties of the random variables or processes, such as expected values, variances, and probabilities of specific events, using the chosen probability models
• Conduct sensitivity analyses to assess the impact of parameter uncertainty or model assumptions on the problem's solution and to identify the most influential factors or sources of uncertainty
• Interpret the results of the probabilistic analysis in the context of the original engineering problem, communicating the findings, recommendations, and limitations of the analysis to stakeholders and decision-makers
• Iterate and refine the probabilistic model as needed, incorporating new data, insights, or feedback to improve the accuracy and reliability of the analysis and to support ongoing decision-making processes
  • Example: Calculate the probability of observing at least 3 defects in a sample of 100 products, given a Poisson distribution with an average of 1.5 defects per 100 products
  • Example: Determine the 95th percentile of the time between equipment failures, modeled by an exponential distribution with a mean of 500 hours, to establish a maintenance schedule that minimizes downtime