Alphabeta MathFinite Probability Spaces and Random Variables — Examples and Counterexamples
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 11 results · all verified · 6 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 5 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
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
Example
A loaded die as a nonuniform finite probability space
Example
Bayes' theorem for choosing one of two urns and observing a colour
Example
Three pairwise-independent events that are not mutually independent
Counterexample
Events with the same probability need not be independent
Counterexample
Uncorrelated finite random variables need not be independent
Counterexample
For dependent variables, E[XY] need not equal E[X]E[Y]
False statement
False: linearity of expectation requires independence
Example
The union bound can be strict for overlapping events
Example
A symmetric two-point distribution attains equality in Chebyshev's inequality
Example
A two-valued random variable attains equality in Markov's inequality
Counterexample
Markov's conclusion can fail without nonnegativity
Sources
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 1.2
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 4.1
- 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
- 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., Theorem 6.2
- 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 8.1 after Example 8.1
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 6.2.2