Expectation of a real random variable on a finite probability space

Depends on

• Real random variables on finite probability spaces and their finite distributions
• Finite sums and finite products, by recursion

Used by

• Expectation agrees with the published finite weighted sum Corollary
• Equal posterior means do not control the full posterior Counterexample
• Expectation equal to 1 does not force a nonnegative integer-valued variable to vanish somewhere Counterexample
• For dependent variables, E[XY] need not equal E[X]E[Y] Counterexample
• Markov's conclusion can fail without nonnegativity Counterexample
• Expected query cost from finite execution prefixes Definition
• Finite expectation-estimation task Definition
• Finite populations and sampling designs Definition
• Finite type-II testing function with explicit orientation Definition
• Reconstruction autoencoder and denoising risk Definition
• The moment generating function M_X(t)=E[e^tX] on a finite probability space Definition
• Two-stage randomized design and strategy-specific effects Definition
• Variance, standard deviation, and covariance on a finite probability space Definition
• A binary Neyman–Pearson boundary randomization Example
• A certainty unit and a census under Horvitz–Thompson estimation Example
• A failed stationary covariance and a valid quadratic variogram Example
• A finite-history conditional mean can be nonlinear Example
• A normalized-pixel objective with a constant representation Example
• A paired contrast with unequal unit-level effects Example
• A rank-deficient universal-kriging design Example
• A singular history covariance can have nonunique coefficients Example
• A symmetric two-point distribution attains equality in Chebyshev's inequality Example
• A two-valued random variable attains equality in Markov's inequality Example
• An atomic sinusoid and an AR(1) process have different spectra Example
• Ordinary kriging can have larger error than simple kriging Example
• Pairwise uncorrelated data can be jointly dependent Example
• Pass-at-one, oracle pass-at-N, and verifier-selected accuracy differ Example
• Positivity failure, near-positivity weights, and an MNAR sensitivity curve Example
• Singular observation covariance in simple kriging Example
• SRSWOR inclusion probabilities, pairwise nonpositive indicator covariance, and the sample mean Example
• A random inner linear code has fewer than one bad word in expectation Lemma
• Cauchy-Schwarz for finite random variables: E[XY]² leE[X²]E[Y²] Lemma
• Conditional expectation constructs the inner code deterministically Lemma
• Conditional expectation on positive-probability events in finite spaces Lemma
• Dyadic positive-mean estimator under an L2 bound Lemma
• Expectation is the sum of each attained value times its probability Lemma
• Finite iid log-likelihood averages concentrate Lemma
• Finite simple-random-sampling moment identities Lemma
• Finite two-stage sampling variance identities Lemma
• Indicators turn event probabilities, intersections, and finite counts into expectations and products Lemma

…and 20 more results.

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 6.1 (standard reference, not scraped)
• M. Bucic, Probabilistic Method, Definition A.5 (standard reference, not scraped)