STATS-2: Statistics for Data Science 2

Topics covered

• Introduction to Multiple Random Variables

  • Review of random variables
  • Two random variables defined on the same sample space
  • Joint probability distributions

• Joint Probability Mass Function (Joint PMF)

  • Definition of joint PMF
  • Properties of joint PMF
  • Calculating probabilities using joint PMF

• Marginal Distributions

  • Marginal PMF
  • Obtaining marginal distributions from joint distributions
  • Marginal probabilities

• Joint Cumulative Distribution Function

  • Joint CDF
  • Properties of joint CDF
  • Calculating probabilities using joint CDF

• Joint and Marginal Distributions

  • Understanding the relationship between joint and marginal distributions
  • Working with probability tables

Expected skills: Calculate joint and marginal probabilities and understand the relationship between two random variables.

Topics covered

• Independence of Random Variables

  • Definition of independence
  • Independence using joint and marginal PMFs
  • Testing independence

• Functions of Random Variables

  • Transformations of random variables
  • Functions of two random variables
  • Distribution of a transformed variable

• Visualizing Functions of Random Variables

  • Geometric interpretation
  • Graphical understanding of transformations

• Functions of Multiple Random Variables

  • Sum of random variables
  • Difference of random variables
  • Other transformations

• Distribution of Functions

  • Calculating probabilities for transformed random variables
  • Understanding the effect of transformations on distributions

Expected skills: Test independence and calculate distributions of functions of random variables.

Topics covered

• Casino Mathematics

  • Expected winnings
  • Fair games
  • Expected value in practical situations

• Expected Value

  • Expectation of discrete random variables
  • Expectation of functions of random variables
  • Linearity of expectation

• Scatter Plots and Spread

  • Visualizing relationships between variables
  • Understanding spread and variation

• Variance and Standard Deviation

  • Variance calculations
  • Standard deviation
  • Properties of variance

• Covariance

  • Definition of covariance
  • Positive and negative covariance
  • Covariance calculations

• Correlation

  • Pearson correlation coefficient
  • Interpretation of correlation
  • Relationship between covariance and correlation

• Inequalities

  • Markov's inequality
  • Chebyshev's inequality
  • Applications of probability bounds

Expected skills: Calculate expectation, variance, covariance and correlation, and use probability inequalities.

Topics covered

• Discrete vs. Continuous Random Variables

  • Difference between discrete and continuous random variables
  • Probability mass function versus probability density function
  • Probability at a single point for continuous variables

• Weight Data

  • Continuous measurements
  • Understanding real-world weight distributions
  • Visualizing data through histograms

• Density Functions

  • Probability density function (PDF)
  • Properties of a PDF
  • Area under the density curve
  • Calculating probabilities over intervals

• Expectations

  • Expected value of continuous random variables
  • Integrating a function against a PDF
  • Variance and other moments for continuous variables

Expected skills: Understand PDFs, calculate interval probabilities using integration and find expectations of continuous random variables.

Topics covered

• Multiple Continuous Random Variables

  • Height and weight data
  • Two continuous random variables
  • Joint probability density functions
  • Marginal distributions
  • Dependence between variables

• Averages of Random Variables

  • Averages of independent observations
  • Behaviour of averages
  • Colab illustration

• Limit Theorems

  • Law of Large Numbers
  • Central Limit Theorem
  • Approximation using normal distributions

• IPL Data and Approximate Distributions

  • IPL data analysis
  • Histograms
  • Approximate probability distributions

• Jointly Gaussian Random Variables

  • Joint Gaussian distributions
  • Dependence and correlation
  • Bivariate normal models

• Probability Models for Data

  • Simple probability models
  • Models based on other distributions
  • Models involving multiple random variables
  • Dependency in probability models
  • Models for IPL powerplay data
  • Building models from data

Expected skills: Work with joint continuous distributions, understand limit theorems, interpret Gaussian models and construct probability models for data.

Topics covered

• Revision topics

  • Revision of Weeks 1–5
  • Joint discrete distributions
  • Independence and functions of random variables
  • Expectation, variance and covariance
  • Continuous random variables
  • Multiple continuous distributions
  • Probability models and limit theorems
  • Practice of earlier assignment concepts

Expected skills: Consolidate the probability and random-variable concepts before moving to estimation and inference.

Topics covered

• Introduction to Statistical Inference

  • Population parameters
  • Sample statistics
  • Statistical inference from sample data

• Point Estimation

  • Estimators and estimates
  • Estimating population mean
  • Estimating population variance
  • Sample mean and sample variance

• Sampling Distributions

  • Distribution of sample statistics
  • Sampling behaviour of estimators
  • Standard error

• Methods of Estimation

  • Method of Moments
  • Maximum Likelihood Estimation (MLE)
  • Likelihood functions
  • Log-likelihood functions

Expected skills: Understand point estimators, calculate estimates from sample data and use basic estimation methods.

Topics covered

• Properties of Estimators

  • Bias and unbiasedness
  • Variance of an estimator
  • Mean squared error
  • Comparing estimators

• Interval Estimation

  • Point estimates versus interval estimates
  • Confidence intervals
  • Confidence level
  • Margin of error

• Confidence Intervals for Population Parameters

  • Confidence intervals for the population mean
  • Confidence intervals when population variance is known
  • Confidence intervals when population variance is unknown
  • Confidence intervals for variance

• Interpretation of Confidence Intervals

  • Repeated sampling interpretation
  • Effect of sample size
  • Effect of confidence level

Expected skills: Understand estimator properties, construct confidence intervals and interpret uncertainty in parameter estimates.

Topics covered

• Introduction to Bayesian Estimation

  • Frequentist versus Bayesian approaches
  • Prior distributions
  • Likelihood functions
  • Posterior distributions

• Bayes' Theorem for Parameter Estimation

  • Prior probability
  • Data likelihood
  • Posterior probability

• Bayesian Estimation

  • Posterior mean
  • Posterior mode
  • Maximum a posteriori (MAP) estimation

• Bayesian Models

  • Beta-Binomial model
  • Normal models, where applicable
  • Updating beliefs using observed data

Expected skills: Understand Bayesian reasoning, calculate posterior distributions in basic models and estimate unknown parameters.

Topics covered

• Introduction to Hypothesis Testing

  • Statistical hypotheses
  • Null hypothesis (H₀)
  • Alternative hypothesis (H₁)
  • One-sided and two-sided tests

• Test Statistics

  • Constructing test statistics
  • Sampling distributions under the null hypothesis

• Significance Level

  • Type I error
  • Type II error
  • Significance level (α)
  • Statistical power

• P-values

  • Meaning of a p-value
  • Calculating and interpreting p-values
  • Decision rules

• Tests for Population Mean

  • Z-tests
  • Test decisions using critical values and p-values

Expected skills: Set up hypotheses, calculate test statistics, interpret p-values and make decisions using a specified significance level.

Topics covered

• Further Hypothesis Testing

  • Tests for population parameters
  • One-sided and two-sided testing
  • Critical regions and rejection rules

• Tests for Population Mean

  • Tests when population variance is unknown
  • T-tests
  • Interpretation of test results

• Tests for Population Variance

  • Chi-square distribution
  • Chi-square tests for variance

• Regression Models

  • Introduction to simple linear regression
  • Relationship between variables
  • Estimating regression coefficients
  • Testing regression parameters
  • Interpreting regression results

Expected skills: Select appropriate tests for mean and variance, interpret results and understand hypothesis testing in simple regression.

Topics covered

• Revision topics

  • Revision of joint probability distributions
  • Independence and transformations
  • Expectations, variance and covariance
  • Continuous and joint continuous distributions
  • Limit theorems and Gaussian models
  • Point and interval estimation
  • Bayesian estimation
  • Hypothesis testing
  • Regression models
  • Practice and revision of the course material

Expected skills: Connect probability, estimation and hypothesis testing concepts and solve comprehensive course-level questions.