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