7  Summary

This crash course has comprehensively covered five essential modules in probability theory and numerical methods, providing a solid foundation for engineering students preparing for examinations:

  1. Discrete Probability Distributions: Explored Probability Mass Functions (PMF), expected values, variances, Binomial distributions for fixed-trial scenarios, Poisson distributions for rare events, and joint PMF for dependent variables.

  2. Continuous Probability Distributions: Covered Probability Density Functions (PDF), Uniform distributions for equal-probability intervals, Exponential distributions for waiting times, Normal distributions with Z-score calculations, and the Central Limit Theorem for approximating sums.

  3. Random Processes: Introduced Wide-Sense Stationary (WSS) processes with constant means and lag-dependent autocorrelations, Power Spectral Density via Fourier transforms, and Poisson processes for modeling random arrivals and inter-arrival times.

  4. Numerical Methods - I: Detailed Newton-Raphson iterative root-finding, finite difference approximations, Lagrange and Newton polynomial interpolation for data estimation, and numerical integration using Trapezoidal and Simpson’s rules.

  5. Numerical Methods - II: Advanced techniques including Gauss elimination for linear systems, Jacobi and Gauss-Seidel iterative solvers, least squares regression for data fitting, Euler, Runge-Kutta method, and the Milne method for solving ordinary differential equations, and Simpson’s 3/8 rule for higher-accuracy integration.

Throughout the modules, emphasis was placed on practical applications, with solved problems demonstrating key theorems and methods, and practice exercises reinforcing understanding. The course equips students with the tools to tackle probability calculations, numerical approximations, and problem-solving strategies essential for exam success.