• Probability notes
  • Printable course contents
    • Slides
    • Printer-friendly pdf files
    • Problems
    • Exams
  • 1 Random experiments
    • 1.1 Events (sets)
    • 1.2 Probability
    • 1.3 Conditional probability
    • 1.4 Bayes’ formula
    • 1.5 Independence
    • 1.6 Combinatorics
  • 2 Discrete random variables
    • 2.1 Definition of random variable
    • 2.2 Discrete r.v.s, probability mass function, and cumulative distribution function
    • 2.3 Mean, variance, and quantiles
    • 2.4 The Bernoulli process
      • 2.4.1 Binomial distribution binom(size,prob)
      • 2.4.2 Geometric (Pascal’s) distribution geom(prob)
      • 2.4.3 Negative Binomial distribution nbinom(size,prob)
    • 2.5 Hypergeometric distribution hyper(m,n,k)
    • 2.6 Poisson distribution pois(lambda)
  • 3 Continuous random variables
    • 3.1 Density mass function and cdf
    • 3.2 Mean, variance, and quantiles
    • 3.3 Uniform distribution
    • 3.4 Transformations of a random variable
    • 3.5 Exponential distribution exp(rate=1)
    • 3.6 Normal distribution
  • 4 Random vectors
    • 4.1 Joint, marginal, and conditional distributions
    • 4.2 Independence
    • 4.3 Transformations of random vectors
    • 4.4 Sums of independent random variables (convolutions)
    • 4.5 Mean vector and covariance matrix
    • 4.6 Multivariate Normal and Multinomial distributions
      • 4.6.1 Multivariate normal distribution mvnorm(mean,sigma)
      • 4.6.2 Multinomial distribution multinom(size,prob)
    • 4.7 Mixtures
    • 4.8 General concept of a random variable
    • 4.9 Random sample
    • 4.10 Order statistics
  • 5 Properties of the expectation
    • 5.1 Expectation and variance of a linear combination of random variables
    • 5.2 Conditional expectation
    • 5.3 Conditional variance
    • 5.4 Moments of a random variable
    • 5.5 The moment generating function
  • 6 Limit Theorems
    • 6.1 Markov and Chebishev inequalities
    • 6.2 Weak LLN (convergence in probability)
    • 6.3 Central Limit Theorem (convergence in distribution)
    • 6.4 Strong LNN (almost sure convergence)
  • References
  • Published with bookdown

Notes for Probability

Notes for Probability

Master in Statistics for Data Science at UC3M

Ignacio Cascos

2019-09-10, v1.0

Printable course contents

Slides

  • Slides of Random experiments
  • Slides of Discrete random variables
  • Slides of Continuous random variables
  • Slides of Random vectors
  • Slides of Properties of the expectation
  • Slides of Limit theorems

Printer-friendly pdf files

  • Random experiments
  • Discrete random variables
  • Continuous random variables
  • Random vectors
  • Properties of the expectation
  • Limit theorems

Problems

  • Random experiments ; with solutions
  • Discrete random variables ; with solutions
  • Continuous random variables ; with solutions
  • Random vectors 1/2 ; with solutions
  • Random vectors 2/2 ; with solutions
  • Properties of the expectation ; with solutions

Exams

  • Final exam, November 2018 ; with solutions
  • Extraordinary exam, June 2019 ; with solutions