Date 
Content of the Lecture 
Assignments/Readings/Notes 
18/07 (Tue) 
 Introduction, course overview and course policies
Descriptive Statistics
 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
 Data summarization: mean and median


21/07 (Fri) 
 Data summarization: mean and median
 Proofs that median minimizes the sum of absolute deviations: with and without using calculus
 Concept of quantile
 Standard deviation and variance, some applications
 Twosided Chebyshev inequality with proof; Oneside Chebyshev inequality (ChebyshevCantelli inequality)
 Concept of correlation coefficient, proof that its value lies from 1 to +1


25/07 (Tue) 
 Correlation coefficient: properties; uncentered correlation coefficient; limitations of correlation coefficient and Anscombe's quartet
 Correlation and causation
 Proof of onesided Chebyshev's inequality
MATLAB/SciLab demo


28/07 (Fri) 
 MATLAB demo: code vectorization, vector and matrix manipulation; graphical plots: plots, surface plots, boxplots, scatterplots; functions from statistics Code snippets
 SciLab demo code
Discrete Probability
 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


1/8 (Tue) 
 Birthday paradox in discrete probability
Random Variables
 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 proof for the former)
 Variance and standard deviation, with alternate expressions
 Markov's and Chebyshev's inequality: with proofs


4/8 (Fri) 
 Weak law of large numbers: proof using Chebyshev's inequality
 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
 Properties of covariance


8/8 (Tue) 
 Concept of conditional PDF, CDF, PMF; conditional expectation and variance with examples
 Concept of moment generating function, two different proof of uniqueness of moment generating function for discrete random variables, properties of momenet generating functions


18/8 (Fri) 
Families of Random Variables
 Bernoulli PMF: mean, median, mode, variance, MGF
 Binomial PMF: relation to Bernoulli PMF, mean, median, mode, variance, plots, MGF, difference between binomial and geometric or negative binomial distribution
 Gaussian (normal) PDF: motivation from the central limit theorem, illustration of central limit theorem


22/8 (Tue) 
 Gaussian PDF: derivation of MGF, median, mode; expression for CDF and its relation to the error function
 Gaussian (normal) PDF: motivation from the central limit theorem, illustration of central limit theorem
 Statement of central limit theorem and its extensions; proof of CLT; application of CLT and its relation to the binomial distribution  de MoivreLaplace theorem (without proof)
 Derivation of PDF of mean of different random variables; Bessel's correction for standard deviation


28/8 (Mon: extra) 
 PDF of sample mean and sample variance of a Gaussian
 Chi square distribution: mean, variance, MGF; and its use towards deriving the PDF of the sample variance of a Gaussian
 Uniform distribution: mean, variance, median, MGF; applications in sampling from a prespecified PMF; application in generating a random permutation of a given set
 Poisson distribution: mean, variance, MGF, mode, addition of Poisson random variables, examples; derivation of Poisson from binomial


29/8 (Tue) 
 Poisson distribution: mean, variance, MGF, mode, addition of Poisson random variables, examples; derivation of Poisson from binomial
 Exponential distribution: relation to Poisson distribution, mean, median, CDF, MGF, memorylessness property, minimum of exponential random variables
 Multinomial PMF  generalization of the binomial, mean vector and covariance matrix for a multinomial random variable, MGF for multinomial


29/8 (Tue: extra lecture) 
Parameter estimation
 Concept of parameter estimation (or parametric PDF/PMF estimation)
 Maximum likelihood estimation (MLE)
 MLE for parameters of Bernoulli, Poisson, Gaussian and uniform distributions
 Least squares line fitting as an MLE problem
 Concept of estimator bias


1/9 (Fri) 
 Concept of estimator bias, variance and MSE, proof that MSE = variance + squared bias
 Examples of biased estimators: variance of Gaussian when mean is unknown, parameter of uniform distribution
 Interval estimates: twosided confidence intervals


5/9 (Tue) 

 Quiz solutions up on moodle

8/9 (Fri) 
 Interval estimates: twosided and onesided confidence intervals
 Confidence interval for mean of a Gaussian with known standard deviation
 Confidence interval for variance of a Gaussian
 Approximate confidence interval for mean of Bernoulli
 Application of MLE  capturerecapture method for counting of animals
 Aside: proof (using MGFs) that the sum of two Gaussian random variables is also Gaussian distributed

