Probability, Statistical and Random Processes

MA25C11:

Probability, Statistical and Random Processes

Course Objective:

To provide a rigorous mathematical foundation in statistical analysis and probability theory required for industrial quality control, semiconductor manufacturing, signal processing, and modern communication systems. The course emphasizes "small sampling theory" and stochastic processes essential for electronics R&D.

Descriptive and Bivariate Statistics: Univariate Measures: Mean, Median, Mode, Variance, and Standard Deviation, Relative Variation: Coefficient of Variation (CV) for comparing component stability., Bivariate Statistics: Covariance and the Correlation Coefficient (r), Linear Regression: Method of Least Squares for sensor calibration and trend analysis

Probability Foundations and Distributions

Probability Theory: Axioms, Conditional Probability, and Bayes’ Theorem (Signal Detection).

Random Variables: Discrete (Binomial, Poisson for shot noise) and Continuous (Uniform, Exponential).

The Normal Distribution: Properties of Gaussian distributions, Central Limit Theorem, and Z-scores.

● Multivariate Gaussian: Joint PDFs, independence, and correlation in vector random variables.

Statistical Inference and Small Sampling

Sampling Distributions: Population vs. Sample; Degrees of Freedom.

Small Sample Tests: Student’s t-distribution (One-sample and Two-sample tests). Variance Analysis: Chi-square distribution and F-distribution for comparing production batches.

Estimation: Confidence Intervals for mean and variance of device parameters.

Hypothesis Testing and Error Metrics

Testing Framework: Null (H0) and Alternative (H1) Hypotheses.

Decision Errors: Type I error (Alpha), Type II error (Beta), and Power of a test.

p-values: Interpretation of p-values in industrial datasheets and medical electronics.

Parametric & Non-Parametric: ANOVA basics and Chi-square Goodness-of-Fit tests.

Stochastic Processes and Signal Noise

Random Processes: Concept of Stationarity (WSS) and Ergodicity.

Correlation Dynamics: Auto-correlation and Cross-correlation functions.

Frequency Domain: Power Spectral Density (PSD) and Wiener-Khinchin Theorem.

LTI Systems: Response of linear circuits to random noise; Introduction to the Kalman Filter.

Weightage: Continuous Assessment: 40%, End Semester Examinations: 60%.

Mandated Activities with weightage: Assignments, MATLAB Activities, Tutorials, Quiz.

Assessment Methodology: Assignments (20%), Solution to application-oriented problems

using software (20%), Solving competitive examinations questions (20%), Internal Examinations (40%).

References:

1 Roy D Yates and David J Goodman, “Probability and Stochastic Processes”, Wiley India, New Delhi, 2021.

2 Douglas C Montgomery and George C Runger, “Applied Statistics and Probability for Engineers”, 7th edition, Wiley India, New Delhi, 2020.

3 Oliver C. Ibe, “Fundamentals of Applied Probability and Random Processes”, 1st IndianReprint, Elsevier, 2014.

4 Saeed Ghahramani, “Fundamentals of Probability with Stochastic Processes”, 5th edition, CRC Press, USA, 2024.

5 Stark. H and Woods J.W, "Probability and Random Processes with Applications to Signal Processing", 3rd Edition, Pearson Education, Asia, 2001.