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