Date 
Content of the Lecture 
Assignments/Readings/Notes 
18/07 (Mon) 
 Introduction, course overview and course policies
 Descriptive statistics: key terminology
 Methods to represent data: frequency tables, bar/line graphs, frequency polygon, piechart
 Concept of frequency and relative frequency
 Cumulative frequency plots
 Interesting examples of histograms of intensity values in an image


21/07 (Thurs) 
 Interesting examples of histograms of intensity values in an image
 Concept of mean, median, mode, percentile, standard deviation and variance with examples
 Mean as minimizer of total squared deviations, median as minimizer of sum of absolute deviations
 Chebyshev's inequality: twosided and onesided with examples


25/07 (Mon) 
 Proof of Chebyshev's inequality: twosided and onesided
 Correlation coefficient: centered and uncentered versions, properties and examples
 Correlation and causation
 A demo of a simple MATLAB program


28/07 (Thurs) 

 Please consult some of the MATLAB tutorials mentioned above on this webpage
 Examples covered in class: matrix and vector operations,
code vectorization, functions for different types of plots and graphs, statistical functions (mean, median, variance, standard deviation)

01/08 (Mon) 
 Discrete probability: sample space, event, composition of events: union, intersection, complement, exclusive or, De Morgan's laws
 Boole's and Bonferroni's inequalities
 Conditional probability, Bayes rule, False Positive Paradox


04/08 (Thurs) 
 Random variable: concept, discrete and continuous random variables
 Probability mass function (pmf), cumulative distribution function (cdf) and probability density function (pdf)
 Expected value for discrete and continuous random variables
 Expected value of a function of a random variable
 The mean and the median as minimizers of squared and absolute losses respectively (with proofs)
 Variance and standard deviation, with alternate expressions
 Markov's and Chebyshev's inequality: with proofs

 Slides
 Read chapter 4 of the textbook

08/08 (Mon) 
 Weak law of large numbers along with proof, statement of strong law of large numbers
 Gambler's fallacy
 Concept of joint PMF, PDF, CDF
 Concept of covariance, concept of mutual independence and pairwise independence
 Concept of moment generating function, two different proof of uniqueness of moment generating function for discrete random variables, properties of momenet generating functions

 Slides
 Read chapter 4 of the textbook

11/08 (Thurs) 
 Concept of conditional PDF, CDF, PMF; conditional expectation and variance with examples
 Bernoulli, binomial and Poisson distributions and their properties: mean, variance, MGF, mode and median (in some cases)


18/08 (Thurs) 
 Gaussian distribution: mean, variance, median, mode, MGF, other properties
 Central limit theorem: statement of theorem, MATLAB code to demo the theorem, and one application


22/08 (Mon) 
 Proof of central limit theorem using the MGF
 de Moivre Laplace theorem  stated without proof
 Distribution of sample mean and sample covariance  chisquare distribution and its MGF for n degrees of freedom, genesis of the chi square distribution for n = 1
 Uniform distribution  mean, median, variance, MGF, application in sampling from arbitrary PMFs


25/08 (Thu) 
 Exponential distribution: motivation, pdf, cdf, mean, variance, MGF, memorylessness
 Multinomial distribution: concept of mean vector and covariance matrix; mean, covariance and MGF of multinomial
 Introduction to hypergeometric distribution


29/08 (Mon) 
 Concept of maximum likelihood estimation
 Maximum likelihood (ML) estimates for parameters of Bernoulli, Poisson, Gaussian and uniform distributions
 Concept of biased estimator and example (ML estimator of the variance of a Gaussian when the mean is also unknown)
 Introduction to the concept of the variance of an estimator

 Slides
 Read sections 7.1, 7.2, 7.7 of the textbook

1/09 (Thu) 
 Bias, variance, mean squared error of an estimator, proof that mean squared error = squared bias + variance; consistency of an estimator
 Derivation of bias, MSE, variance for two different estimators of the parameter of a uniform distribution
 Concept of confidence interval  onesided and twosided, examples for mean of a Gaussian with known variance, variance of a Gaussian, mean of a Bernoulli (approximate)

