Probability and Statistics syllabus

Basics of set Theory: Introduction to sets and algebra of sets, Random Experiment, Sample Space, Events, Complementary Events, Union and Intersection of Two Events, Difference Events, Exhaustive Events, Mutually Exclusive Events, Equally Likely Events, Independent Events. Probability Theory: Mathematical & Statistical definition of Probability, Need of probability theory in Data science, Axiomatic definition of probability, Addition Theorem, Multiplication Theorem, Theorems of Probability, Conditional Probability, Inverse Probability, Joint Probability, Total Probability and Bayes Theorem.

Case Study: Use of probability in real-life situations, like weather forecasting, sports betting, sales forecasting etc

Introduction to Statistics: Introduction, Origin and Development and scope of Statistics, Population and Sample, Sampling –Introduction, Types of Sampling, Purposive Sampling, Random Sampling, Simple Sampling, Stratified Sampling, Parameter and Statistic, Sampling Distribution Sampling With and Without Replacement, Population Parameters, Sample Statistics. Introduction, Arithmetic Mean, Simple and weighted mean for raw data, Discrete frequency distribution, Continuous frequency distribution, Properties of A.M.,Merits & Demerits of A.M. Median, Mode for raw data, Merits and demerits of Median and Mode.

Case Study : Create measures of central tendency for a real-life example dataset, such as the payroll dataset or titanic dataset.

Case study of sampling for any real-world problem like exit poll statistics

Measures of Dispersion, Skewness and Kurtosis: Dispersion, Characteristics for an Ideal Measure of Dispersion, Measures of Dispersion, Range, Quartile Deviation, Mean Deviation, Standard Deviation and Root Mean Square Deviation, Coefficient of Dispersion, Coefficient of Variation, Skewness, Kurtosis. Correlation and Regression : Bivariate Distribution, Scatter diagrams, Correlation, Karl Pearson’s coefficient of correlation, Rank correlation, Regression, Regression Coefficients, Lines of Regression.

Case study: Create measures of dispersion for a real-life example dataset like students dataset, iris detection etc.

Random Variables and Distribution Functions: Random Variable, Distribution Function, Properties of Distribution Function, Discrete Random Variable, Probability Mass Function, Discrete Distribution Function, Continuous Random Variable, Probability Density Function, moment generating function, median and quantiles, Markov inequality, Chebyshev’s inequality. Theoretical Discrete Distributions: Binomial and multinomial distributions, Bernoulli Distribution, Mean Deviation about Mean of Binomial Distribution, Mode of Binomial Distribution, Additive Property of Binomial Distribution, Characteristic Function of Binomial Distribution, Cumulants of Binomial Distribution, Poisson distribution, Uniform distribution, Exponential distribution, Gaussian distribution, Log-normal distribution, Chi-square distribution.

Case study : Use Binomial distribution for the problem of reducing errors by vendors who process credit-card applications for a large credit-card bank etc.

Hypothesis and Testing of Hypothesis: Introduction, Statistical Hypothesis (Simple and-Composite), Test of a Statistical Hypothesis, Null Hypothesis, Alternative Hypothesis, Critical Region, Two Types of Errors, level of Significance, Power of the Test. Steps in Solving Testing of Hypothesis Problem, Optimum Tests Under Different Situations, Most Powerful Test (MP Test), Uniformly Most Powerful Test, Likelihood Ratio Test, Properties of Likelihood Ratio Test. Neyman-Pearson Fundamental Lemma, Test for the Mean of a Normal Population, Test for the Equality of Means of Two Normal Populations, Test for the Variance of a Normal Population, Test for Equality of Variances of two Normal Populations, Non-parametric Methods, Advantages and Disadvantages of Non-parametric Methods.

Case study: Study hypothesis testing for any examples like to determine whether the female proportion of the adult population is high or any similar example