Probability I
Skills
Every skill in the course, and how you are doing on each.
Probability spaces, axioms, and rigorous counting
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Normalization
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Monotonicity
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Exactly one
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Inclusion exclusion
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Telescoping series
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Countable additivity
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Continuity of probability
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Countable union bound
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Complement counting
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Adjacency
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Double counting
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Sampling without replacement
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Surjections
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Permutation symmetry
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Null events
150 more topics with one problem
- Aggregation
- Equally likely outcomes
- Sample space
- Event operations
- Finite probability law
- Disjoint decomposition
- Probability axioms
- Event partition
- Two event inclusion exclusion
- De morgan laws
- Event algebra
- Truth table
- Disjointization
- Equality condition
- Union bound
- Divisibility counting
- Bonferroni bound
- Finite model
- Sharp bounds
- Countable sample space
- Tail event
- Limits of laws
- Uniformity
- Proof
- Infinitely often
- Summability
- Digit strings
- Local versus global constraints
- Multiplication principle
- Choosing positions
- Words
- Block method
- Forbidden adjacency
- Permutations
- Labelled preimages
- Multiset permutations
- Circular permutations
- Rotations
- Cyclic gap
- Rotational symmetry
- Unequal orbits
- Committees with roles
- Uniform model
- Hypergeometric count
- Ordered versus unordered
- Lower bounds
- Stars and bars
- Occupancy models
- Sample space choice
- Unequal weights
- Bounded compositions
- Capacity constraints
- Bijection
- Line versus circle
- Nonadjacent selections
- Subset selection
- Vandermonde identity
- Hockey stick identity
- Largest element
- Binomial identity
- Parity bijection
- Subsets
- Equal block sizes
- Labelled versus unlabelled
- Set partitions
- Occupancy shapes
- Derangements
- Exactly one fixed point
- Multinomial coefficient
- Polynomial expansion
- Weighted counting
- First appearance
- Relative order
- Lattice paths
- Obstacles
- Ballot paths
- Prefix constraint
- Reflection bijection
- Binary strings
- Counting identity
- Recurrence counting
- Distinct symbol count
- Model comparison
- Random subsets
- Staged sampling
- Finite partition
- Generated sigma algebra
- Observable events
- Probability distance
- Symmetric difference
- Infinite events
- Sharp dependence bounds
- Finite uniform refinement
- Rational probabilities
- Sample space construction
- Forbidden pairs
- Perfect matchings
- Symmetry factors
- Binary words
- Positive compositions
- Runs
- At least k events
- Intersection bound
- Finite intersections
- Infinite product
- Continuous probability
- Countability
- Countable null union
- Continuity bounds
- Event limits
- Liminf limsup
- Collision probability
- Extremal bound
- Nonuniform sampling
- Conditional output
- Decision paths
- Shuffle bias
- Conditional length
- Directed distance
- Permutation cycles
- Geometric probability
- Random line
- Slice weighting
- Overlapping events
- Random tournament
- Transitive order
- Degree constraints
- Labelled trees
- Prufer codes
- Finite product
- First collision
- Sequential counting
- Component decomposition
- Connectivity recurrence
- Random graph
- Eulerian counts
- Insertion recurrence
- Permutation descents
- Binary matrices
- Linear independence
- Sequential basis count
- Conditional shapes
- Lattice geometry
- Random quadrilaterals
- Conditional preference
- Parking process
- Rotational counting
- Labelled assignments
- Orthogonal partitions
- Shared blocks
Conditional probability and independence
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Conditional probability
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Independence
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Bayes rule
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Conditional measure
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Compatibility
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First step analysis
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Increasing events
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Almost sure equality
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Optimal decisions
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Conditional independence
119 more topics with one problem
- Restricted sample space
- Reversed conditioning
- Complement events
- Disjoint events
- Chain rule
- Sequential conditioning
- Source attribution
- Total probability
- Base rate
- Posterior threshold
- Likelihood ratio
- Posterior odds
- Sequential evidence
- Host policy
- Informed selection
- Observation likelihood
- Common target population
- Mixture reversal
- Stratification
- Coarse information
- Conditional rates
- Independence from partition
- Restricted universe
- Conditioning stability
- Information effect
- Sharp bound
- Joint table
- Reverse conditioning
- Conditional union
- Overlapping information
- Retained information
- Successive conditioning
- Countable partition
- Geometric series
- Posterior tail
- Posterior bounds
- Uncertain rates
- Evidence normalization
- Multiple hypotheses
- Posterior direction
- Degenerate case
- Mixture identification
- Limiting conditioning
- Null event
- Observation specification
- Eventual outcome
- Repeated trials
- Alternating turns
- First player advantage
- Restart equation
- Boundary conditions
- First step equations
- Hitting probability
- Overlapping patterns
- Pattern race
- State selection
- Coarse reports
- Likelihood proportionality
- Stopping information
- Depletion
- Hidden composition
- Posterior prediction
- Conditional cancellation
- Independent blocks
- Reliability events
- Conditional limits
- Nonmonotone probabilities
- Independence boundaries
- Logical implication
- Joint likelihood
- Sharp posterior bounds
- Unspecified dependence
- Loss of information
- Noisy channel
- Recursive conditioning
- Conditional bayes
- Ignorable selection
- Report retention
- Posterior classification
- Probability of error
- Randomized decisions
- Posterior uncertainty
- Refined information
- Conditional transition
- Stationary chain
- Time reversal
- Coarsened observation
- Remaining positions
- Stopping report
- Record selection
- Threshold strategy
- Unavailable offers
- Fair bit extraction
- Paired trials
- Unknown bias
- Interval observation
- Inverse weighting
- Length biased sampling
- Inductive invariant
- Reservoir sampling
- Uniform subsets
- Common mass
- Maximal coupling
- Mismatch probability
- Positive association
- Product measure
- Exchangeability
- Reinforced urn
- Absorption
- Conditional survival
- Quasi stationary law
- Bridge reliability
- Conditional network
- Pivotal edge
- Jump chain bias
- Metropolis rule
- Rejection stays
- Grid crossings
- Random orientation
Random variables and distribution functions
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Preimage counting
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Atoms
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Mixed distribution
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Density normalization
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Symmetry
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CDF continuity
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Quantiles
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Geometric distribution
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Poisson distribution
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Lognormal distribution
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Truncation
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Distribution identification
118 more topics with one problem
- Probability mass function
- Random variable
- Cumulative distribution
- Endpoint conventions
- CDF validity
- Right continuity
- CDF jump
- Density limitation
- Density interpretation
- Null points
- Probability versus density
- CDF integration
- Finite information
- Inverse events
- Measurability
- Countable atoms
- Jump mass
- Discrete transform
- Generalized quantile
- Inverse sampling
- Continuous inverse
- Probability transform
- Decreasing transformation
- Inverse exponential
- Support
- Changing support
- Multiple preimages
- Nonmonotone transformation
- Discrete hazard
- Distribution reconstruction
- Survival probability
- Memorylessness
- Support convention
- Continuous memorylessness
- Exponential distribution
- Survival equation
- Complement tail
- Probability mass derivation
- Adjacent mass ratio
- Binomial mode
- Tied maxima
- Binomial assumptions
- Dependent trials
- Feasible support
- Hypergeometric distribution
- Negative binomial
- Stopping count
- Terminal success
- Parity probability
- Conditional mass function
- Shifted distribution
- Zero truncation
- Density scaling
- Normal distribution
- Standardization
- Location scale identification
- Normal quantiles
- Folding
- Half normal
- Jacobian factor
- Transformed quantiles
- Gamma density
- Integration by parts
- Poisson tail identity
- Beta density
- Integer parameters
- Integral recursion
- Boundary atom
- Censoring
- Discretization
- Floor transformation
- Geometric connection
- Median definition
- Nonunique quantiles
- Dense thresholds
- Lifetime quantiles
- Nonconstant hazard
- Survival function
- Memorylessness failure
- Mixture lifetimes
- Posterior survival
- Bounded density
- Concentration bound
- Interquartile range
- Rejection sampling
- Repeated acceptance
- Density mixture
- Geometric sum
- Latent count
- CDF range
- Dense atoms
- Dyadic support
- Incomplete decades
- Leading digits
- Random rescaling
- Biased digits
- Periodic thresholds
- Self similar law
- Broken stick
- Largest spacing
- Piecewise CDF
- CDF comparison
- Quantile coupling
- Stochastic order
- Moment nonuniqueness
- Perturbed density
- Discrete arcsine law
- Last return
- Random walk
- Hidden atom
- Maximum observation
- Partial identification
- Finite moment identification
- Interpolation
- Support size
- Asymmetric laplace
- Exponential difference
- Sign magnitude
Expectation, moments, and transforms
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Sharpness
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Indicators
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Linearity
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Unbiased estimation
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Equality cases
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Exponential moments
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Markov inequality
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Convexity
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Cauchy schwarz
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Factorial moments
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Independent copy
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Iterated expectation
96 more topics with one problem
- Finite distribution
- Functions of random variables
- Weighted averages
- Continuous expectation
- Lotus
- Null sets
- Integrability
- Positive negative parts
- Undefined expectation
- Countable union
- Nonnegative expectation
- Integer valued variables
- Tail sum
- Capped loss
- Heavy tails
- Tail integral
- Records
- Inclusion probabilities
- Stopped sums
- Centering
- Squared error
- Variance
- Bounded variance
- Sharp inequalities
- Nonnegative integration
- Tail moments
- Moment existence
- Strict thresholds
- Chebyshev inequality
- Second moment
- One sided inequality
- Optimization
- Reciprocal payoff
- Supporting line
- Square integrability
- Triangle inequality
- Lower tail bound
- Second moment method
- Independent samples
- Sample variance
- Parity
- Probability generating function
- Size bias
- Convergence domain
- Gamma distribution
- Moment generating function
- Atom at zero
- Laplace transform
- Mixture distribution
- Characteristic function
- Complex expectation
- Uniform distribution
- Chernoff bound
- Degenerate distributions
- Moment constraints
- Transform validity
- Cumulant generating function
- Exponential tilting
- Absolute loss
- Medians
- Displacement
- Pattern comparison
- Self overlap
- Covariance identity
- Monotone functions
- Exact expectation
- Generating integrals
- Reciprocal binomial moments
- Branching process
- Extinction
- Absolute distance
- CDF integral
- Overlapping indicators
- Random graphs
- Triangle variance
- Generating polynomial
- Insertion process
- Permutation cycle count
- Coding length
- Entropy bound
- Logarithmic inequality
- Convex order
- Finite hinge representation
- Stop loss transform
- Convex order incomparability
- Equal moments
- Threshold payoffs
- Expectation versus path
- Leave one out jensen
- Sample mean convex order
- Bernstein differences
- Common random environment
- Mixed binomial convex order
- Independence boundary
- Pairwise probability balancing
- Poisson binomial convex order
Joint distributions and conditional expectation
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Predictive probability
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Binomial distribution
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Nonidentical trials
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Conditional distribution
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Conditional density
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Multinomial distribution
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Marginal of multinomial
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Aggregation of categories
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Conditional multinomial
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Conditioning on a count
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Indicator method
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Multinomial covariance
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Conditional expectation
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Finite joint distributions
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Conditional densities
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Order statistics
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Density factorization
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Jacobian
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Joint CDF
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Overlapping sums
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Density transformation
139 more topics with one problem
- Multinomial coefficients
- Variance of a sum
- Weighted totals
- Binomial splitting
- Category grouping
- Multinomial theorem
- Disjoint cases
- Covariance of indicators
- Occupancy counts
- Multivariate hypergeometric
- Mixture distributions
- Mutual independence
- Pairwise independence
- Deterministic transformations
- Zero covariance
- Conditional covariance
- Random sample size
- Total variance
- Marginal densities
- Triangular support
- Integration regions
- Joint densities
- Range distribution
- Change of variables
- Joint mass factorization
- Poisson splitting
- Discrete transformations
- Parity constraints
- Discrete support
- Finite differences
- Continuous marginals
- Singular joint law
- Discrete convolution
- Continuous convolution
- Moving support
- Conditioning and integration
- Integrable singularity
- Product distribution
- Conditional scale
- Ratio distribution
- Covariance of sums
- Independent increments
- Translated support
- Continuous mixture
- Conditional allocation
- Odds weights
- Attainable correlation
- Bernoulli marginals
- Coupling bounds
- Bernoulli correlation
- Extremal coupling
- Integer variance bound
- Conditioning induced dependence
- Covariance sign
- Event selection
- Conditional mean prediction
- Least squares
- Nonlinear dependence
- Event conditioning
- Finite triangular law
- Conditional moments
- Deterministic threshold
- Conditional normal law
- Gaussian independence
- Orthogonal residual
- Joint density
- Transformed support
- Uncorrelated dependence
- Mixed poisson model
- Overdispersion
- Total covariance
- Degenerate parameters
- Zero variance
- Count covariance
- Distinct trial indicators
- Conditional occupancy
- Multinomial support
- Report selection
- Finite population covariance
- Fixed margin table
- Hypergeometric law
- Covariance
- Transformed discrete law
- Support geometry
- Couplings
- Dependence
- Tail probability
- Competing risks
- Exponential race
- Winner independence
- Fractional part
- Mixed coordinates
- Conditional versions
- Local conditioning
- Normal ratio
- Polar coordinates
- Radial independence
- Copula density
- Correlation range
- Parameter validity
- Correlation feasibility
- Gaussian construction
- Singular endpoints
- Bivariate poisson
- Common shocks
- Joint generating function
- Biased relabelling
- Induced dependence
- Ordered pairs
- Brier score
- Calibration
- Reliability resolution decomposition
- Age residual joint law
- Discrete inspection
- Length biased observation
- Age residual independence
- Geometric characterization
- Support boundary
- Additive equality case
- Finite product decomposition
- Variance tensorization
- Covariance decay
- Independent refresh noise
- Interaction spectrum
- Conditional block count
- Fixed size negative association
- Monotone coupling
- Continuous conditioning
- Dependence diagnostics
- Triangular support integration
- Conditional moment audit
- Conditional normalization
- Weighted triangle integration
- Bivariate change of variables
- Factorization and dependence
- Transformed support analysis
- Posterior moment checks
- Predictive density normalization
- Support reversal
Modes of convergence, laws of large numbers, and CLT
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Almost sure convergence
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Convergence in probability
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Mean convergence
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Weak law
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Strong law
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Central limit theorem
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Asymptotic independence
94 more topics with one problem
- Moment convergence
- Almost sure failure
- Moving indicators
- Convergence in distribution
- Coupling
- Subsequences
- Convergence implications
- CDF sandwich
- Constant limits
- Distributional convergence
- Dependent event recurrence
- Kochen stone
- Second moment lower bound
- Borel cantelli
- Independent events
- Infinite occurrence
- Subsequence principle
- Bounded convergence
- Error splitting
- Finite sample bound
- Sample average
- Dependence counterexample
- Uncorrelated variables
- Finite first moment
- Fourth moment
- Summable probabilities
- Ratio estimator
- Retained observations
- Empirical distribution
- Finite support
- Total variation
- Approximation vs bound
- Exponential sums
- Binomial normal approximation
- Continuity correction
- Lattice distribution
- Inconsistent scale
- Normal mixture limit
- Delta method
- Slutsky theorem
- Square root transform
- Joint convergence
- Product normal limit
- Second order product
- Poisson limit
- Rare events
- Endpoint spacings
- Joint extreme limits
- Integrable tails
- Maximal observation
- Characteristic function proof
- Dominated convergence
- Diverging count variance
- Rare event threshold
- Triangular binomial CLT
- Continuous mapping
- Deterministic sequences
- Discontinuity
- Products of limits
- Slutsky mechanism
- Tightness
- Fatou lemma
- Uniform moment bound
- Approximation error
- Quantitative CLT
- Third absolute moment
- Flat CDF
- Nonunique median
- Oscillating order statistic
- Geometric exponential limit
- Rescaled waiting time
- Boundary terms
- Dependent CLT
- Coupon collection
- Factorial moment limit
- Gumbel threshold
- Gaussian maxima
- Logarithmic scaling
- Tail bounds
- Concentration
- Moving windows
- Summable errors
- Absent first moment
- Cauchy stability
- Characteristic functions
- Binomial counts
- Cauchy median
- Quantile CLT
- Geometric normal sum
- Laplace limit
- Random count fluctuations
- Coloured fixed points
- Joint poisson limit
- Mixed factorial moments