Alphabeta MathFinite Probability Spaces and Random Variables
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✓ 19 results · all verified · 6 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- limsup, liminf, and Subsequential Limits
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
2 · Logical flowchart
3 · Items
Definition
Finite probability spaces, outcome weights, events, and event probabilities
Lemma
Normalization, nonnegativity, monotonicity, complements, and differences in a finite probability space
Theorem
Probability is additive on every finite pairwise-disjoint family of events
Corollary
Two-event inclusion-exclusion: ℙ(A∪ B)=ℙ(A)+ℙ(B)-ℙ(A∩ B)
Theorem
The finite union bound
Definition
Conditional probability ℙ(A∣ B) for ℙ(B)>0
Theorem
The multiplication rule and finite chain rule for conditional probability
Theorem
The law of total probability for a finite partition
Theorem
Bayes' theorem over a finite partition
Definition
Independent events, pairwise independence, and mutual independence of a finite family
Theorem
Two positive-probability events are independent exactly when conditioning on either leaves the other's probability unchanged
Lemma
Mutual independence is inherited by subfamilies and by replacing events with complements
Definition
The uniform probability space on a nonempty finite set
Definition
The finite product of finite probability spaces
Theorem
Product weights normalize, and coordinate events are mutually independent
Definition
Real random variables on finite probability spaces and their finite distributions
Definition
Expectation of a real random variable on a finite probability space
Lemma
Expectation is the sum of each attained value times its probability
Theorem
Expectation is linear for every finite family of random variables, without any independence hypothesis
Theorem
Expectation preserves pointwise order and lies between the minimum and maximum attained values
Definition
The indicator random variable of an event
Lemma
Indicators turn event probabilities, intersections, and finite counts into expectations and products
Definition
Pairwise and mutual independence of finite-valued random variables
Theorem
A finite family of events is mutually independent exactly when its indicators are mutually independent
Definition
Variance, standard deviation, and covariance on a finite probability space
Lemma
Var(X)=E[X²]-E[X]² and Cov(X,Y)=E[XY]-E[X]E[Y]
Definition
Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials
Theorem
Markov's inequality on a finite probability space
Theorem
Chebyshev's inequality on a finite probability space
Lemma
Cauchy-Schwarz for finite random variables: E[XY]² leE[X²]E[Y²]
Theorem
The finite second-moment bound ℙ(X≠0)geE[X]²/E[X²] when E[X²]>0
Corollary
ℙ(X=0)leVar(X)/E[X]² whenever E[X]≠0
Sources
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 1.2
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Sections 1.3.2-1.3.3
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 1.1
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 1.2
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.3.2
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 1.4
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.3.3
- J. Matousek and J. Vondrak, The Probabilistic Method, Lemma 1.1.3
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 6.2.1
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 4.1
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.4
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., equation (4.2)
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 1.3 and equation (4.3)
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.4.2
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Bayes' Formula
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.4.3
- J. Matousek and J. Vondrak, The Probabilistic Method, Definition 1.1.4
- M. Bucic, Probabilistic Method, Definitions A.14-A.15
- J. Matousek and J. Vondrak, The Probabilistic Method, Section 1.1
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.4.1
- M. Bucic, Probabilistic Method, Example A.2
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Definitions 4.3-4.5
- J. Matousek and J. Vondrak, The Probabilistic Method, Definition 1.1.6
- M. Bucic, Probabilistic Method, Definition A.4
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 6.1
- M. Bucic, Probabilistic Method, Definition A.5
- M. Bucic, Probabilistic Method, Appendix A
- J. Matousek and J. Vondrak, The Probabilistic Method, Lemma 3.1.1
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 6.2
- J. Matousek and J. Vondrak, The Probabilistic Method, Definition 3.1.2
- M. Bucic, Probabilistic Method, Section 2.1
- J. Matousek and J. Vondrak, The Probabilistic Method, Lemma 3.1.3
- J. Matousek and J. Vondrak, The Probabilistic Method, Definition 1.1.8
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Sections 1.4.1 and 3.1.5
- J. Matousek and J. Vondrak, The Probabilistic Method, Sections 1.1 and 3.1
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 6.2
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 5.3.1
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 5.1
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 3.1.5
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 6.2.2
- J. Matousek and J. Vondrak, The Probabilistic Method, Theorem 4.1.1
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 8.1
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 6.2.4
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Section 3.3