Date 
Content of the Lecture 
Assignments/Readings/Notes 
18/07 (Wed) 
 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


20/7 (Fri) 
 Data summarization: mean and median
 Proofs that median minimizes the sum of absolute deviations: with and without using calculus
 Concept of quantile/percentile
 Standard deviation and variance, some applications
 Twosided Chebyshev inequality with proof; Oneside Chebyshev inequality (ChebyshevCantelli inequality)
 Concept of correlation coefficient and formula for it

Slides: Descriptive statistics
Readings: section 2.1, 2.2, 2.3, 2.4, 2.6 from the textbook by Sheldon Ross
Noncalculus proof to show that the median minimizes the sum of absolute deviations

25/7 (Wed) 
 Correlation coefficient: properties; uncentered correlation coefficient; limitations of correlation coefficient and Anscombe's quartet
 Correlation and causation
 Proof of onesided Chebyshev's inequality


27/7 (Fri) 
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 (Wed) 
 Conditional probability, Bayes rule, False Positive Paradox
 Birthday paradox
 Independent and mutually exclusive events
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


3/8 (Fri) 
 Proof: The median as minimizer of absolute loss respectively
 Markov and Chebyshev's inequality  with proof
 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 (Wed) 
 Covariance: properties, correlation versus independence
 Concept of moment generating function, two different proofs of uniqueness of moment generating function for discrete random variables, properties of moment generating functions
 PDF/PMF of sum of random variables
 Concept of conditional PDF, CDF, PMF; conditional expectation and variance with examples


10/8 (Fri) 
Families of Random Variables
 Concept of 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 distribution
 Gaussian (normal) PDF: motivation from the central limit theorem, illustration of central limit theorem


17/8 (Fri) 
 Gaussian (normal) PDF: motivation from the central limit theorem, illustration of central limit theorem
 Derivation of mean, variance, MGF, median, mode; CDF of a Gaussian and its relations to error functions; probability of a Gaussian random variable to have values between mu +/ k sigma.
 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); one application of the CLT; relation between CLT and the law of large numbers


18/8 (Sat) 
 Gaussian tail bounds
 Distribution of the sample mean and the sample variance, Bessel's correction;
 Chisquared distribution  definition, genesis, MGF, properties; use of a chisquare distribution toward defining the PDF of the sample variance
 Uniform distribution: mean, variance, median, MGF; applications in sampling from a prespecified PMF; application in generating a random permutation of a given set


20/8 (Mon) 
 Poisson distribution: mean, variance, MGF, mode, addition of Poisson random variables, examples; derivation of Poisson from binomial
 Relation between Poisson and Gaussian distributions, examples
 Multinomial PMF  generalization of the binomial, mean vector and covariance matrix for a multinomial random variable, MGF for multinomial


24/8 (Fri) 
 Exponential distribution: mean, median, MGF, variance, property of memorylessness, minimum of exponential random variables
 Overview of hypergeometric distribution  capturerecapture problem in ecology
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
 Concept of estimator bias
 Least squares line fitting as an MLE problem


29/8 (Wed) 
 Concept of estimator bias, mean squared error, variance
 Estimators for interval of uniform distribution: example of bias
 Least squares line fitting as an MLE problem
 Concept of twosided confidence interval
 Confidence interval for mean of a Gaussian with known standard deviation


31/8 (Fri) 
Quiz


5/9 (Wed) 
 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, using the hypergeometric distribution

