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4.4. Probability and Statistics (PST)

This area covers the mathematical modeling of uncertainty and the analysis of engineering data. It ranges from the axiomatic foundations of probability to statistical inference methods essential for quality control, reliability engineering, and data-driven design.

Knowledge Area (KA)	Core Tier1	Core Tier2
4.4.1 Mathematical Statistics and Estimation Theory	1	1
4.4.2 Time Series Analysis	1	1
4.4.3 Probability Axioms and Random Variables	Elective
4.4.4 Stochastic Processes and Markov Chains	Elective

4.4.1. PST/Mathematical Statistics and Estimation Theory  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Rigorous methods for estimating population parameters from sample data in engineering experiments and quality control.
Topics:
Core

• Point estimation: Method of Moments and Maximum Likelihood Estimation
• Construction and interpretation of confidence intervals in engineering tests
• Hypothesis testing for engineering quality control and process validation
• Linear and nonlinear regression for engineering data fitting
• Introduction to Design of Experiments (DoE) for engineering optimization

Learning Outcomes:
Core:

1. Derive Maximum Likelihood Estimators for parameters of engineering failure models [Usage]
2. Verify the statistical significance of engineering test results using hypothesis testing [Assessment]
3. Construct confidence intervals for key process parameters from experimental data [Usage]
4. Apply linear regression to model and predict engineering performance from measured data [Assessment]

4.4.2. PST/Time Series Analysis  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Stationary processes, ARMA models, spectral analysis, state-space models, and forecasting.
Topics:
Core

• Stationarity, autocovariance function, and the autocorrelation function (ACF)
• ARMA models: identification, estimation (Yule-Walker, MLE), and diagnostics
• Spectral density, the periodogram, and Wiener-Khinchin theorem
• State-space models and the Kalman filter
• ARIMA models, seasonal adjustment, and multi-step forecasting

Learning Outcomes:
Core:

1. Identify stationarity and determine the order of an ARMA model from ACF and PACF plots [Familiarity]
2. Fit ARIMA models to time series data, validate residuals, and produce forecasts [Usage]
3. Apply the Kalman filter to estimate hidden states in a linear Gaussian state-space model [Assessment]

4.4.3. PST/Probability Axioms and Random Variables ↑ Back to top

Foundations of probability theory including sample spaces, probability measures, and random variables with engineering applications in reliability and uncertainty quantification.
Topics:
Core

• Sample spaces, events, and the Kolmogorov axioms
• Discrete and continuous random variables and their distribution functions
• Expectation, variance, and moments of engineering random variables
• Common engineering distributions: Normal, Exponential, Poisson, and Weibull
• Joint distributions, covariance, and multivariate normal distribution

Learning Outcomes:
Core:

1. Calculate probabilities for discrete and continuous distributions relevant to reliability engineering [Usage]
2. Compute the mean, variance, and standard deviation of random variables describing engineering quantities [Usage]
3. Evaluate the probability density function for transformed random variables and apply it to failure analysis [Assessment]
4. Model multivariate engineering uncertainties using joint probability distributions [Usage]

4.4.4. PST/Stochastic Processes and Markov Chains ↑ Back to top

Models for engineering systems that evolve randomly over time, including queuing, reliability, and signal noise models.
Topics:
Core

• Discrete-time Markov chains, transition matrices, and steady-state analysis
• Poisson processes and their application in reliability and queuing engineering
• Stationarity and ergodicity in engineering random signals

Non Core

• Brownian motion and its application in vibration and financial engineering

Learning Outcomes:
Core:

1. Compute stationary distributions for Markov chain models of engineering system states [Assessment]
2. Model arrival processes in queuing systems using the Poisson distribution [Usage]
3. Explain the concept of power spectral density for stationary random engineering signals [Familiarity]

NonCore:

1. Simulate a simple Brownian motion model and interpret its variance growth over time [Usage]