/STAT-134-Probability-Theory

STAT-134 Introduction to Probability Theory

STAT-134 Probability Theory

University of California, Berkeley (Summer 2019)

Author: John-Michael Laurel

Remarks: These documents assume understanding of basic probabilistic knowledge such as independence, conditioning, Bayes' rule, joint distributions, et cetera, counting methods (combinatorics), and set theory e.g. deMorgan's laws. Happy Studying ☕

Continuous Treatise

version 1.5 (08-12-2019)

  • Section 8.3 Rayleigh Distribution
    • fixed the pdf and cdf, had the 1/2 in the wrong place

version 1.4 (08-11-2019)

Big update today. Material Slate is no more, say hello to its successor: Black Slate.

Changes and Additions

  • Section 5.1 Uniform Distribution
    • retracted fix (from version 1.3) on construction for expectation, actually is (a+b)/2
  • Section 7 Poisson Arrival Process
    • new additions (section 7.4 and 7.5) for merging and thinning Poisson processes, along with diagrams
    • fixed indices for inter-arrival times, if there are r arrivals, there are r inter-arrival times not r-1
    • fixed parameter for inter-arrival times should be lambda not lambda times t
  • Section 9 Operations - removed density for Z = X-Y, it's really just another case of Z = X+Y
  • Section 10 Conditional Expectation
    • Law of Iterated Expectation (Towering Rule)
    • other properties
  • Section 11 Co-Variance
    • defintion
    • bi-linearity
    • variance of a sum of exchangeable random variables
    • variance covariance matrix
  • Section 12 Correlation
    • definition
    • properties
    • sampling an entire population
  • Section 13 Standard Bi-Variate Normal
    • construction
    • properties

version 1.3 (08-07-2019)

  • Section 6 Gamma Distribution
    • added namedDistribution "Gamma" with respective parameters
  • Section 5.1 Uniform Distribution
    • fixed construction for expectation should be (b-a)/2 not (a+b)/2

version 1.2 (08-07-2019)

  • Section 8.2 Linear Combinations of Normal
    • changed "rotational symmetry" to "spherical symmetry" since we're technically in n dimensions
    • appended footnote to specify rotational symmetry when we're dealing with a sum of two independent standard normals
  • Section 8.3 Rayleigh Distribution
    • Corrected named distribution on the square root of a sum of two squared standard normals, should be Rayleigh not N(0,1).
  • Added Section 9 Operations

version 1.0 (08-06-2019)

  • Dark Mode is here! I made two variations:
    • Material Slate (Comfortable Contrast)
    • Jet Black (High Contrast)
  • Content
    • Continuous Random Variables
    • Change of Variable for Densities
    • Cumulative Distribution Functions (CDF)
      • minimums and maximums
      • using CDF to compute expectation
    • Distributions
      • Uniform
      • Exponential
        • memoryless property
        • competing exponentials
      • Gamma
        • Gamma function
        • Diagram: Poisson Arrival Process
      • Beta
        • Order Statistics
        • Uniform Order Statistics
      • Normal
        • linear combinations of normal
        • Rayleigh
        • sum of squared normals

Discrete Treatise

version 2.1 (08-31-2019)

  • Updated document blurb
  • Section 6 Distributions and Their Relationships
    • renamed to "Discrete Distributions Schematic"
    • appended a "+" on the last term b {a,a+1,...,+b}
    • adjusted wording for Uniform distribution
    • added \sum without replacement for arrow connecting Bernoulli(p) to Hypergeometric(n,N,G)
  • Section 2.1.1
    • cleaned up some wording
  • Section 10.2.1
    • adjusted footnote 6

version 2.0 (08-06-2019)

  • Updated document description
  • Added metadata to PDF file
  • Section 3.4 Multinomial Distribution
    • added index starting value for N

Version 1.3 (07-22-2019)

  • updates headers styles: small caps -> boldface
  • added section: Law of Large Numbers
  • Section 3.8 Poisson Distribution
    • added sums of Poissons is Poisson
  • Section 7.2 Chebyshev's Inequality
    • update some notation
  • Section 10.2.1 Variance of a Sum of Indicators
    • added a matrix containing indicator terms in product expansion of a sum of indicators
    • cleaned up paragraph structures

Version 1.2 (07-18-2019)

  • Section 3.5 Geometric Distribution
    • updated synopsis and description
  • Schematic
    • segregated "Uniform Distribution" and updated its parameters to match parameters in section 3.1
  • added a "Techniques and Important Problems" section

Version 1.1 (07-17-2019)

  • Additions
    • tail bounds Markov and Chebyshev
    • variance of a sum of dependent indicators
    • Craps Principle
  • Section-5 Central Limit Theorem
    • rule of thumb for CLT approximation instead of {range of X} should be {range of S_n}
    • added standardization asterisk for a and b

Version 1 (07-16-2019)

  • brief treatise on discrete probability
  • random variables and names distributions
  • overview of distributions
    • probability mass function
    • expectation
    • variance
  • schematic