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Expectation

In probability theory and statistics, expectation, also known as expected value, is a fundamental concept that represents the long-run average value of a random variable over infinitely many independent repetitions of an experiment. For a discrete random variable X with possible values x_i and probabilities p_i, it is defined by the formula E[X] = \sum_i x_i p_i; for a continuous random variable with probability density function f(x), it is E[X] = \int_{-\infty}^{\infty} x f(x) \, dx.^[1]^[2]^[3] The notion of expectation emerged in the mid-17th century amid efforts to resolve problems in games of chance, particularly the "problem of points," which concerned fair division of stakes in interrupted games. Blaise Pascal and Pierre de Fermat developed early ideas through their correspondence in 1654, laying groundwork for probability, while Christiaan Huygens formalized the concept in his 1657 treatise De ratiociniis in ludo aleae, where he introduced mathematical expectation as a way to value gambles based on weighted outcomes.^[4]^[5] This work built on earlier intuitive uses in gambling and insurance but marked the first rigorous mathematical treatment, influencing subsequent developments by Jacob Bernoulli and others in the late 17th and 18th centuries.^[4] Expectation possesses key properties that make it a powerful tool in analysis. It is linear, so for constants a and b and random variables X and Y, E[aX + bY] = aE[X] + bE[Y], allowing simplification of complex expressions without knowing the joint distribution. It is also monotonic: if X \geq Y almost surely, then E[X] \geq E[Y] when the expectations exist. Additionally, for indicator random variables (taking values 0 or 1), the expectation equals the probability of the event occurring, E[I_A] = P(A). These properties hold under mild conditions, such as finite integrability, and extend to conditional expectation, which is the best predictor of a variable given partial information.^[6]^[7]^[8] Beyond theory, expectation finds wide applications in decision-making, finance, and risk assessment. In insurance, it helps calculate fair premiums by estimating average claims, as when an insurer charges the expected payout plus a margin to cover costs. In investments, expected value guides portfolio choices by weighting potential returns by probabilities, aiding in assessments of profitability under uncertainty. It also underpins algorithms in machine learning for optimizing expected rewards and in operations research for evaluating strategies in stochastic environments.^[9]^[10]^[11]

General concept

Definition

Expectation refers to the act or state of expecting or anticipating something to occur, typically involving a belief or mental outlook directed toward future events.^[12] This concept encompasses a forward-looking attitude where individuals form ideas about probable outcomes based on prior experiences, information, or hopes.^[13] In essence, it represents a psychological posture of awaiting, which can range from confident foresight to tentative projection.^[14] At its core, expectation involves several key components: a belief oriented toward the future, which may be grounded in reality or influenced by subjective factors, and the potential for emotional responses depending on whether the anticipated event materializes.^[15] If expectations are unmet, this can lead to feelings such as disappointment or surprise, highlighting the emotional stakes tied to such anticipations.^[16] The term underscores the human tendency to project current knowledge or desires onto uncertain futures, shaping behavior and decision-making in daily life.^[17] The word "expectation" derives from the Latin exspectatio, which means "a looking out for" or "awaiting," stemming from exspectare ("to look out for"), a combination of ex- ("out") and spectare ("to look").^[18] This etymological root entered English in the 16th century via Middle French expectation, evolving to denote both the process of anticipation and the object anticipated.^[19] Common examples illustrate expectation in everyday contexts, such as anticipating rain after consulting a weather forecast, which influences preparations like carrying an umbrella, or holding social expectations in relationships, where one might foresee mutual support based on established norms.^[13] These scenarios demonstrate how expectations guide actions and interactions without delving into technical applications, such as the expected value in probability theory, which formalizes anticipation of outcomes.^[15]

Etymology and usage

The term "expectation" derives from the Latin exspectātiō, the noun form of exspectō, meaning "to look out for" or "to await," composed of ex- (thoroughly) and spectō (to look at or watch).^[19] It entered the English language in the early 16th century, borrowed partly from Middle French expectation (14th century) and directly from Latin, with its first known use in 1523 in the writings of Thomas Wolsey.^[18]^[12] This linguistic root emphasizes a forward-looking gaze, evolving from literal observation to mental projection of future outcomes. In contemporary usage, "expectation" shares synonyms such as anticipation and expectancy, which often carry positive connotations of hopeful foresight, while terms like apprehension or foreboding evoke negative undertones of anxiety or dread.^[20] These variations highlight subtle differences: positive expectations imply optimism and preparation, whereas negative ones suggest fear of disappointment, influencing how the word shapes interpersonal and narrative dynamics. Philosophically, expectation functions as a belief about probable future events, bridging anticipation with cognitive projection.^[21] Culturally, expectation permeates literature and idiomatic expressions, underscoring human aspirations and disillusionments. Charles Dickens' 1861 novel Great Expectations thematically dissects the burdens of social and personal ambitions through protagonist Pip's journey, illustrating how inflated hopes can lead to moral reckoning and growth.^[22] Common phrases like "beyond expectation" capture the surprise of exceeding anticipated outcomes, often invoked in motivational or reflective contexts to denote delightful unpredictability.^[23] In modern non-technical domains, expectation plays a pivotal role in business and education. In business, customer expectations encompass anticipated service quality, personalization, and responsiveness, directly impacting loyalty and satisfaction when met or surpassed.^[24] Similarly, in education, learner expectations involve students' projections of supportive environments and clear goals, fostering engagement when aligned with teaching practices that emphasize effort and equity.^[25]

Mathematics and statistics

Expected value

In probability theory, the expected value of a random variable represents the long-run average value of repetitions of the experiment that would be achieved with unlimited trials, serving as a measure of central tendency for the distribution.^[26] Also known as the mathematical expectation, it quantifies the average outcome weighted by the probabilities of each possible result.^[27] This concept was introduced by Blaise Pascal in 1654 through his correspondence with Pierre de Fermat, initially developed to resolve gambling problems such as the "problem of points," where players divide stakes fairly based on interrupted games.^[28] For a discrete random variable X taking values x_i with probabilities p_i where \sum p_i = 1, the expected value is calculated as

E[X] = \sum_i x_i p_i.

This formula weights each outcome by its likelihood, providing the mean under the probability mass function. In the continuous case, for a random variable X with probability density function f(x), the expected value is

E[X] = \int_{-\infty}^{\infty} x f(x) \, dx,

integrating over the support of the distribution to account for the continuum of outcomes.^[3] Expected value exhibits key properties that facilitate its use in analysis. Linearity holds regardless of dependence between variables, such that for constants a, b and random variables X, Y, E[aX + bY] = a E[X] + b E[Y], enabling decomposition of complex expectations into simpler components.^[29] Additionally, if X is non-negative (i.e., P(X \geq 0) = 1), then E[X] \geq 0, reflecting the preservation of non-negativity under probabilistic averaging.^[3] These properties underpin applications in risk assessment, where expected value helps quantify potential losses or gains under uncertainty, such as in insurance pricing or project evaluation.^[30] In decision-making, it supports choices by comparing average outcomes across options, as seen in portfolio optimization or policy analysis.^[31] The law of total expectation, also known as the law of iterated expectations or the tower property, states that for random variables X and Y, the expected value of X equals the expected value of the conditional expectation of X given Y:

E[X] = E[E[X \mid Y]].

This theorem allows decomposition of expectations over conditional distributions, facilitating computations in complex probabilistic models. For the discrete case, the derivation proceeds by expressing the unconditional expectation as a sum over possible values of Y:

E[X] = \sum_y E[X \mid Y = y] P(Y = y) = E[E[X \mid Y]],

where the outer expectation is taken with respect to the distribution of Y. For continuous variables, the sum is replaced by an integral:

E[X] = \int E[X \mid Y = y] f_Y(y) \, dy,

yielding the same result.^[32] Linearity of expectation asserts that for any random variables X_1, \dots, X_n (not necessarily independent) and constants a_1, \dots, a_n,

E\left[\sum_{i=1}^n a_i X_i\right] = \sum_{i=1}^n a_i E[X_i].

This property holds regardless of dependence between the X_i, making it a powerful tool for analyzing sums of random variables, such as in indicator variable methods for counting problems. The proof follows directly from the definition of expectation: for the discrete case,

E\left[\sum_i a_i X_i\right] = \sum_\omega P(\omega) \sum_i a_i X_i(\omega) = \sum_i a_i \sum_\omega P(\omega) X_i(\omega) = \sum_i a_i E[X_i],

with the interchange justified by finite sums; the continuous case uses integrals analogously.^[33] The expectation-maximization (EM) algorithm provides an iterative procedure for finding maximum likelihood estimates in models with latent variables or incomplete data. It alternates between an E-step, which computes the expected value of the log-likelihood of the complete data (treating latent variables as missing), given the current parameter estimates, and an M-step, which maximizes this expected log-likelihood to update the parameters. This process monotonically increases the observed-data likelihood and converges to a local maximum under standard conditions. The algorithm was developed by Arthur P. Dempster, Nan M. Laird, and Donald B. Rubin in 1977.^[34] Key applications of the EM algorithm include parameter estimation in hidden Markov models (HMMs) and clustering via Gaussian mixture models. In HMMs, the E-step employs the forward-backward algorithm to compute expected state occupancies and transitions from observations, while the M-step updates transition probabilities a_{ij} as ratios of expected transitions to expected state visits, and similarly for emission probabilities. For Gaussian mixtures, the E-step calculates posterior probabilities of component assignments, and the M-step updates mixing coefficients \pi_k, means \mu_k, and covariances as weighted averages of the data, enabling soft clustering of points into probabilistic components. These uses underpin tasks like speech recognition and density estimation.^[35]

Physics

Vacuum expectation value

In quantum field theory, the vacuum expectation value (VEV) of a field operator \phi is defined as \langle 0 | \phi | 0 \rangle, where |0\rangle is the vacuum state representing the ground state of the system. This quantity captures the average value of the field in the absence of excitations and serves as a fundamental object for computing observable effects in the theory. For scalar fields, the VEV vanishes in symmetric vacua but becomes non-zero when the theory exhibits spontaneous symmetry breaking, shifting the minimum of the potential away from the origin.^[36] The VEV concept emerged in the 1960s amid efforts to unify the weak and electromagnetic forces through electroweak theory. In 1964, François Englert and Robert Brout demonstrated that a non-zero VEV for a scalar field could induce masses for gauge bosons via spontaneous symmetry breaking in a gauge-invariant manner. Peter Higgs independently proposed a similar mechanism that same year, applying it to a U(1) gauge model where the VEV generates vector boson masses without producing observable massless scalars. Gerald Guralnik, Carl Hagen, and Tom Kibble concurrently formalized the approach using quantum field operator methods, emphasizing its compatibility with relativistic quantum theories. These works built on earlier ideas from superconductivity and chiral symmetry breaking, establishing the VEV as key to mass generation in particle physics.^[37] Within the Higgs mechanism of the Standard Model, the VEV of the Higgs doublet field \Phi breaks the electroweak SU(2)_L × U(1)_Y symmetry, endowing the W^\pm and Z bosons with mass while leaving the photon massless. The tree-level Higgs potential takes the form V(\Phi) = -\mu^2 |\Phi|^2 + \lambda |\Phi|^4, with \mu^2 > 0 and \lambda > 0 ensuring a stable minimum. Minimization yields the VEV magnitude v = \sqrt{\mu^2 / \lambda}, around which the field fluctuates to produce the observed Higgs boson. Fermion masses arise from Yukawa couplings to this VEV, proportional to the respective coupling strengths.^[37] The Higgs VEV is experimentally constrained to v \approx 246 GeV, derived from the relation v = (\sqrt{2} G_F)^{-1/2} using the precisely measured Fermi constant G_F.^[38] This scale sets the electroweak interaction strength and unifies weak processes with the observed muon decay rate. Quantum corrections to the VEV are computed perturbatively by expanding around the tree-level minimum or via path integral methods that integrate out loop fluctuations to obtain the effective potential. In the path integral formalism, the one-loop Coleman-Weinberg contribution refines the potential, ensuring gauge invariance and renormalization of the VEV.^[39] These calculations confirm the tree-level value to high precision within the Standard Model. The non-zero VEV drives spontaneous symmetry breaking, where degenerate vacua allow the ground state to select a specific direction, violating the original symmetry manifest in the Lagrangian. For a continuous symmetry with n broken generators, this produces n massless Goldstone bosons in global theories; however, in the gauged electroweak sector, these bosons are "eaten" by the gauge fields, manifesting as their longitudinal components and enabling massive vector propagation.^[37] This framework resolves the unitarity issues in high-energy weak scattering and underpins the consistency of the Standard Model.

Expectation in quantum mechanics

In quantum mechanics, the expectation value of an observable represented by a Hermitian operator \hat{A} in a given state |\psi\rangle is defined as \langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle, which provides the average outcome of measurements of that observable on an ensemble of identically prepared systems. This formulation arises from the probabilistic interpretation of the wave function, where |\langle \psi | \psi \rangle|^2 = 1 ensures normalization, and the expectation value is real-valued due to the hermiticity of \hat{A}. The concept was developed in the 1920s during the formulation of quantum mechanics. Max Born introduced the probabilistic interpretation of the wave function in 1926, linking it to expectation values for predicting measurement outcomes, while Werner Heisenberg's matrix mechanics in 1925 used non-commuting operators to compute averages in a discrete basis. Erwin Schrödinger's wave mechanics, published in 1926, extended this to continuous representations, showing that expectation values satisfy classical-like equations under the time-dependent Schrödinger equation. Time evolution of expectation values is governed by the Ehrenfest theorem, which states that for position \langle \hat{x} \rangle and momentum \langle \hat{p} \rangle = -i\hbar \frac{\partial}{\partial x}, the derivatives follow \frac{d}{dt} \langle \hat{x} \rangle = \frac{\langle \hat{p} \rangle}{m} and \frac{d}{dt} \langle \hat{p} \rangle = -\left\langle \frac{\partial V}{\partial x} \right\rangle, bridging quantum and classical dynamics for expectation values. For example, in one dimension, the position expectation is \langle x \rangle = \int_{-\infty}^{\infty} \psi^*(x) \, x \, \psi(x) \, dx, illustrating how the wave function's spatial distribution determines the average location. The Heisenberg uncertainty principle relates expectation values to fluctuations via variances: \Delta A \Delta B \geq \frac{1}{2} \left| \langle [\hat{A}, \hat{B}] \rangle \right|, where \Delta A = \sqrt{\langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2} quantifies the spread around the mean, with the commutator [\hat{A}, \hat{B}] linking non-commutativity to minimal uncertainty. This principle, derived by Heisenberg in 1927 and rigorously proven by Kennard in the same year, underscores that expectation values alone do not fully capture quantum indeterminacy. Applications of expectation values include predicting measurement statistics and analyzing wave packet dynamics, such as in the time evolution of a Gaussian wave packet where \langle x \rangle follows classical trajectories while the width \Delta x spreads due to dispersion. In quantum optics, for instance, expectation values of field operators yield photon number averages in coherent states, essential for laser physics.

Psychology and philosophy

Expectancy effects

In psychology, expectancy effects refer to the ways in which an individual's expectations about events, behaviors, or outcomes can influence perception, cognition, and actual results, often through subtle interpersonal or intrapersonal mechanisms. These effects emerged as a key area of study in the 1960s, building on earlier concepts of self-fulfilling prophecies—where beliefs lead to behaviors that confirm those beliefs—but distinguishing themselves by focusing on unconscious biases and non-volitional responses rather than deliberate actions.^[40]^[41] The observer-expectancy effect, also known as the experimenter-expectancy effect, occurs when a researcher's or observer's preconceptions subtly bias the data collection or participant responses, leading to results that align with those expectations. Pioneering work by Robert Rosenthal in the 1960s demonstrated this through experiments where researchers' beliefs about subjects—such as rats' intelligence or students' potential—influenced outcomes via unintentional cues like tone or attention.^[42] For instance, in a 1963 study with Kermit Fode, experimenters expecting "bright" rats to navigate mazes faster provided more encouragement, resulting in superior performance compared to those labeled "dull," highlighting how expectations shape observed behavior without conscious intent. This effect extends beyond labs to real-world settings, underscoring the need for blind procedures in research. A prominent application is the Pygmalion effect, where higher expectations from authority figures enhance subordinates' performance, creating a positive self-fulfilling dynamic. Rosenthal and Lenore Jacobson's 1968 study in an elementary school randomly informed teachers that certain students were "intellectual bloomers," leading those students to show significant IQ gains over the year due to teachers' increased engagement and positive reinforcement. Empirical evidence from military contexts, such as a 1982 study by Dov Eden and Giora Ravid on Israeli Defense Forces trainees, further supports this: those labeled as high-potential by instructors outperformed peers in boot camp performance.^[43] In education, meta-analyses confirm small but consistent gains in student achievement when teachers hold elevated expectations, often mediated by differentiated instruction and feedback. The placebo effect exemplifies how positive expectations can produce tangible physiological improvements, independent of active treatment. Individuals anticipating symptom relief from inert substances often experience reduced pain or enhanced well-being, with mechanisms involving endorphin release and classical conditioning from prior therapeutic associations.^[44] Brain imaging studies reveal placebo responses activate opioid pathways similar to analgesics, while conditioning—pairing cues like pill shapes with relief—amplifies these via dopamine-mediated reward circuits.^[45] Response expectancies, as theorized by Irving Kirsch in 1985, describe how anticipated non-volitional reactions to stimuli directly elicit those experiences, bridging suggestion and behavior. For example, expecting anxiety in social situations can trigger physiological symptoms like increased heart rate, shaping the encounter accordingly; experimental manipulations altering these expectancies reduce reported distress without changing objective threats. This framework unifies placebo and related effects under expectancy as a core mediator, distinct from broader self-fulfilling prophecies by emphasizing automatic, belief-driven responses.^[46]

Philosophical perspectives

In epistemology, expectation refers to a rational belief about the probability of future events, grounded in available evidence and logical coherence. Philosophers distinguish epistemic expectations as justified anticipations that align with probabilistic reasoning, contrasting them with irrational ones that deviate from evidence-based norms. For instance, rational expectations update in response to new information through processes like Bayesian conditionalization, where prior beliefs are revised proportionally to the likelihood of observed data, ensuring coherence in degrees of belief.^[47] Irrational expectations, by contrast, persist despite contradictory evidence, often leading to epistemic akrasia or weakness of will in belief formation.^[48] Philosophical theories of well-being highlight expectations as central to subjective happiness, particularly through the lens of hedonic adaptation, where individuals adjust their anticipations to align with actual circumstances, thereby stabilizing emotional equilibrium. This adaptation process influences eudaimonic and hedonic accounts of flourishing, as unmet expectations can disrupt perceived well-being, while adaptive recalibration fosters resilience. In normative terms, such dynamics challenge simplistic hedonistic views by underscoring how contextual factors, including social support and coping strategies, modulate the impact of expectations on overall life satisfaction.^[49] Ethically, expectations underpin moral duties, especially in contractual obligations and interpersonal relations, where they impose reciprocal anticipations of fulfillment. In Kantian deontology, dutiful anticipation arises from the categorical imperative, requiring agents to act in ways that universalize their maxims, thereby respecting others' rational autonomy and creating enforceable moral expectations in promises and agreements. This framework views failure to meet such expectations not merely as a breach but as a violation of the good will, emphasizing intention over consequences.^[50] Key thinkers have shaped these perspectives: Aristotle conceptualized hope (elpis) as a forward-looking expectation tied to courage and vulnerability, where good hope (euelpis) emerges from past experiences of fear and safety, integrating it into virtuous character formation. In modern existentialism, Jean-Paul Sartre linked unmet expectations to bad faith (mauvaise foi), a self-deceptive denial of freedom wherein individuals cling to rigid roles or anticipations to evade responsibility for their choices.^[51]^[52] Philosophical tools for eliciting expectations in decision theory include proper scoring rules, which incentivize honest revelation of subjective probabilities by rewarding accurate probabilistic forecasts, as developed in axiomatic frameworks for rational choice. These methods, rooted in von Neumann-Morgenstern utility theory, assess expectations by querying agents on lotteries or hypothetical scenarios, revealing inconsistencies and promoting coherence without relying on empirical observation.^[53]

Arts and entertainment

Music works

Girl's Day's debut full-length album Expectation (Korean: Gidae), released on March 14, 2013, marked a pivotal shift for the group from their earlier cute image to a more mature, sexy concept in K-pop. Produced primarily by Nam Ki-sang, who had crafted their prior hits, the album features 14 tracks, including the title track "Expectation (기대해)," which showcases bold choreography and themes of romantic anticipation, alongside others like "I Don't Mind," "Easy Go," and "Female President" (a later single from the era). It propelled Girl's Day to mainstream success, with the title track earning acclaim for its vocal delivery and contributing to the group's rise in the K-pop industry during 2013.^[54] Keith Jarrett's Expectations, a double album released in October 1972 on Columbia Records, represents his ambitious foray into jazz fusion and big-band arrangements, recorded over three days in April 1972 at Columbia Studio E in New York City. Featuring 11 extended improvisational pieces with contributions from a 17-piece ensemble including strings and brass on select tracks, the album blends post-bop, avant-garde jazz, and rock elements, reflecting Jarrett's evolving compositional style during his early career transition. Originally conceived as a solo piano project, it became a landmark in Jarrett's discography for its expansive, thematic exploration of musical possibility and expectation in improvisation.^[55] The New Zealand rock band Dance Exponents' Expectations, released in May 1985 on Mushroom Records, captured the band's pop-rock energy during their breakthrough period, peaking at number 7 on the New Zealand Albums Chart and maintaining a presence for eight weeks. The album includes 10 tracks such as "You're So Strong," which reached number 11 on the Australian Singles Chart, "Science Fiction," and "My Love for You," blending jangly guitars with introspective lyrics on desire and uncertainty. Achieving double platinum status in New Zealand, it solidified the band's status as a national act, with its title evoking the tension between hopes and realities in relationships.^[56] Tame Impala's "Expectation," a psychedelic rock single released on December 3, 2010, emerged from the recording sessions for their debut album Innerspeaker (2009–2010) at Wave House in Western Australia, later included on the 2020 Collector's Edition. Clocking in at 6:02, the track—written and performed primarily by Kevin Parker—delves into disillusionment with love and the weight of unmet hopes through swirling guitars and introspective lyrics like "Everything you ever told me could have been a lie." Its dreamy structure builds tension to mirror emotional anticipation, influencing Tame Impala's signature psych-rock sound.^[57] D.O. (Do Kyung-soo of EXO)'s second EP Expectation (Korean: Gidae), released on September 18, 2023, by SM Entertainment, comprises seven tracks emphasizing emotional vulnerability and human connection, with the R&B-infused title track "Somebody" urging listeners to transcend preconceptions. Key songs include "Wonder (기적)," the pre-release "I Do (별 떨어진다)," "Lost," "Ordinary Days (내일의 우리)," "The View," and a Performance Team version of "Somebody," showcasing D.O.'s versatile vocals across pop and contemporary R&B styles. Debuting at number 3 on the Circle Album Chart with first-week sales of over 85,000 copies; the title track "Somebody" garnered over 2.065 million streams on MelOn in its first day, it ranked among the top K-pop albums of 2023 for its introspective take on everyday expectations.^[58]) In classical music, the waltz "Expectation" (Russian: Ozidanie, or "Waiting"), composed around 1901 by Russian musician Herold Kittler (1847–1916), evokes a sense of longing through its lilting 3/4 rhythm and melodic swells, often performed by folk orchestras like the Russian Folk Orchestra under Vladimir Fedoseyev. This 19th–early 20th-century piece, rooted in Russian ballroom traditions, uses subtle harmonic tensions to build anticipatory mood, influencing later interpretations in klezmer and neoclassical repertoires.^[59] These works often draw on the concept of expectation as a thematic core, paralleling Eugene Narmour's 1990 implication-realization model, which posits that melodic structures generate listener anticipation through bottom-up perceptual principles like interval size and prosodic rules, creating emotional tension and resolution in music cognition.^[60] In lyrics and forms, titles like Expectation frequently explore anticipation's dual nature—hopeful buildup in Girl's Day's upbeat hooks or Jarrett's improvisational arcs, versus disillusionment in Tame Impala's psych-folk haze—resonating culturally by mirroring personal and relational uncertainties across genres.

Television productions

One of the earliest television productions exploring themes of anticipation and expectation is the 1959 episode "In Lonely Expectation" from the CBS anthology series Playhouse 90. Aired on April 2, 1959, this dramatic play, written by Mayo Simon and directed by Franklin Schaffner, centers on a woman who converts a house into a shelter for unwed mothers, focusing on a teenage girl's dilemma in deciding whether to keep her unborn child amid societal pressures and personal uncertainties.^[61]^[62] The episode exemplifies mid-20th-century live television anthologies that delved into motifs of expectation, often through intimate character studies of moral and emotional waiting, as seen in the golden age of U.S. broadcast drama.^[63] In the modern era, the independent production company Expectation Entertainment has become a prominent entity in unscripted and scripted television formats that incorporate elements of anticipation and expectation. Founded in 2017 by former ITV Director of Television Peter Fincham and ex-Endemol Shine Group President Tim Hincks, the London-based company quickly grew into one of the UK's most influential indies, producing content for major broadcasters like BBC, Channel 4, and Amazon Prime Video.^[64]^[65] Notable among its unscripted output is The Grand Tour, where Expectation handled production for seasons 4 through 6 (2019–2023), featuring road-trip specials that built viewer suspense through unpredictable global adventures involving hosts Jeremy Clarkson, Richard Hammond, and James May.^[66] Expectation's portfolio also includes innovative game shows and chat formats emphasizing real-time recall and surprise, aligning with contemporary unscripted trends that heighten audience engagement through immediate expectations. For instance, I Literally Just Told You, a Channel 4 quiz show hosted by Jimmy Carr and created by Richard Bacon, premiered in December 2021 and challenges contestants to remember details just revealed, with subsequent series in 2022 and 2023 drawing strong young adult viewership—up 34% in volume for 16–34s in its debut.^[67]^[68] Similarly, The Big Narstie Show, a BAFTA-winning late-night entertainment series co-hosted by grime artist Big Narstie and comedian Mo Gilligan, aired its first season on Channel 4 in 2018 and ran through 2022, blending celebrity interviews, sketches, and live music to create anarchic, expectation-defying moments that boosted 16–24 viewership by 94% in its pilot.^[69]^[70] On the scripted side, Expectation produced the BBC Three comedy-drama In My Skin (series 2, 2020), written by Kayleigh Llewellyn, which portrays protagonist Bethan Gwyndaf navigating high school pressures, family secrets surrounding her mother's bipolar disorder, and personal ambitions as a budding writer, underscoring themes of unmet expectations in relationships and self-identity.^[71] Additionally, the company produced the 2025 BBC One documentary Sharon & Ozzy Osbourne: Coming Home, an intimate portrait of the Osbourne family's relocation after 25 years in Los Angeles, capturing Ozzy's health struggles and reflections on legacy amid evolving family dynamics.^[72]^[73] These works highlight Expectation's role in blending narrative tension with real-life unpredictability across genres.

References

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The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable.
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the act or state of looking forward or anticipating. Synonyms: trust, hope, anticipation, expectancy. an expectant mental attitude. a high pitch of expectation.
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The moral theme of Great Expectations is quite simple: affection, loyalty, and conscience are more important than social advancement, wealth, and class.
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ExpectedValue
The expected value or expectation of some quantity X (written E[X] or sometimes just EX) is just the average of all possible values of X weighted by their ...<|control11|><|separator|>
[28]
[PDF] FERMAT AND PASCAL ON PROBABILITY - University of York
The problem was proposed to Pascal and Fermat, probably in 1654, by the Chevalier de Méré, a gambler who is said to have had unusual ability “even for the ...
[29]
Properties of the expected value | Rules and formulae - StatLect
Summary of the properties of the expected value operator, with explanations, proofs, examples and solved exercises.Scalar multiplication of a... · Addition of a constant matrix...
[30]
Expected Value Analysis (Economic Risk Analysis) | EME 460
The expected value is defined as the difference between expected profits and expected costs. Expected profit is the probability of receiving a certain profit ...
[31]
Gauging a Project's Expected Value Using Decision Analysis - PMI
This article examines the process of gauging a project's expected value using decision analysis--also known as risk analysis--to forecast the project outcomes.
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[PDF] Conditional Expectation
May 6, 2020 · funky fool, expectation. ... The law of total expectation states that: E[E[X|Y]] = E[X]. What?! How is that a thing? Check out this proof:.
[33]
[PDF] 3.2.1 Linearity of Expectation
Right now, the only way you've learned to compute expectation is by first computing the PMF of a random variable pX(k) and using the formula E [X] = Pk∈ΩX.
[34]
Maximum Likelihood from Incomplete Data Via the EM Algorithm
A broadly applicable algorithm for computing maximum likelihood estimates from incomplete data is presented at various levels of generality.
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[PDF] A Gentle Tutorial of the EM Algorithm and its Application to ...
In this section, we derive the EM algorithm for finding the maximum-likelihood estimate of the parameters of a hidden Markov model given a set of observed ...
[36]
Zeta-regularized vacuum expectation values - AIP Publishing
Sep 10, 2019 · In a quantum field theory (QFT), vacuum expectation values are fundamental objects. As expectation values of observables, they allow for ...
[37]
Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism (history)
Jan 7, 2009 · But of course its most significant application was to the development of the unified electroweak theory by Weinberg (1967) and Salam (1968), ...
[38]
[PDF] 11. Status of Higgs Boson Physics - Particle Data Group
Dec 1, 2023 · A Higgs boson mass of about 125 GeV allows to explore the Higgs boson couplings to many SM particles. In particular the dominant decay modes ...
[39]
A Supersymmetry Primer
### Summary of VEV Calculation in the Standard Model (SM) from "A Supersymmetry Primer" (hep-ph/9709356)
[40]
Self-fulfilling prophecies: A theoretical and integrative review.
Classroom self-fulfilling prophecies are reviewed in terms of 3 sequential stages: (a) Teachers develop expectations, (b) teachers treat students differently.<|control11|><|separator|>
[41]
A History of the Controversy Over Claims that Teacher Expectancy ...
The 1968 publication of the Rosenthal and Jacobson's Pygmalion in the Classroom offered the optimistic message that raising teachers' expectations of their ...Missing: observer- | Show results with:observer-
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[PDF] WHAT YOU EXPECT IS WHAT YOU GET Rosenthal, R., SL ...
This threat to the validity of a psychological experiment is called the experimenter expectancy effect. Robert Rosenthal, a leading researcher on this ...Missing: observer- | Show results with:observer-<|control11|><|separator|>
[43]
Mechanisms of the placebo effect and of conditioning - PubMed
Dopamine and the endorphins have been clearly shown to be mediators of placebo effects. Brain imaging has demonstrated that placebos can mimic the effect of ...
[44]
Neurobiological Mechanisms of the Placebo Effect
Nov 9, 2005 · Placebo administration (psychosocial context) might reduce pain through opioid and/or non-opioid mechanisms via expectations and/or conditioning ...
[45]
Response Expectancy and the Placebo Effect - PubMed
In this chapter, I review basic tenets of response expectancy theory (Kirsch, 1985), beginning with the important distinction between response expectancies ...
[46]
Bayesian Updating When What You Learn Might Be False ...
Feb 20, 2021 · The standard arguments for Bayesian Conditionalization assume that whenever I become certain of something, it is true. Most people would reject this assumption.
[47]
Epistemic Rationality and the Value of Truth - Duke University Press
Oct 1, 2024 · In this sense of the word, a belief state is irrational when it is out of line with the believer's evidence, whatever favored outcomes it might ...
[48]
The many faces of hedonic adaptation
### Abstract
[49]
[PDF] Kant on "Why Must I Keep My Promise?"
The thesis of this Article is that Kant, in his discussion of contractual obligations, is not relying on traditional moral or philosophical ideas about contract ...
[50]
Full article: Hoping-well: Aristotle's phenomenology of elpis
Oct 8, 2020 · Aristotle tries to solve the riddle of future-directedness and luck-awareness by offering an account of what he calls 'good hope' or hoping-well.
[51]
[PDF] Hope in Ancient Greek Philosophy - PhilArchive
So, Aristotle notes that. “Confidence [is] the expectation [ἐλπίς] associated with an appearance of safety being nearby... . We feel it if we can take steps ...
[52]
Elicitation of Personal Probabilities and Expectations - jstor
Proper scoring rules, i.e., devices of a certain class for eliciting a person's prob- abilities and other expectations, are studied, mainly theoretically ...
[53]
Girl's Day Strip Cute Image With First Full-Length Album 'Expectation'
Mar 19, 2013 · An in-depth look at Girl's Day first full-length album, "Expectation," and what the accomplishment shows.
[54]
Expectations - Keith Jarrett | Album - AllMusic
Rating 8.6/10 (114) Expectations by Keith Jarrett released in 1972. Find album reviews, track lists, credits, awards and more at AllMusic.Missing: fusion | Show results with:fusion
[55]
Dance Exponents - Mushroom Music
The album was a huge success staying in the NZ Album chart for nearly a year, selling double platinum in the process ‚Äì a rare feat for a new New Zealand act ...Missing: tracklist significance
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Video: Tame Impala: "Expectation" | Pitchfork
Dec 2, 2010 · In their video for "Expectation", a track from the BNM'ed 2010 album Innerspeaker, Aussie psych-rockers Tame Impala rock out in the middle ...
[57]
The 25 Best K-Pop Albums of 2023 - Billboard
Dec 29, 2023 · D.O., Expectation. Trending on ... title track encourages listeners to break out of preconceived expectations and connect as humans.
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Old Russian Waltzes, Vladimir Fedoseyev - Qobuz
Expectation. Buy track 00:03:51. Vladimir Fedoseyev, MainArtist - Herold Kittler, Composer ... Waltz. Buy ... Waltz in D Minor. Buy track 00:04:35. Vladimir ...
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[PDF] The Implication Realization Model. - IU ScholarWorks
Eugene Narmour. The Analysis and Cognition of Basic Melodic. Structures: The Implication Realization Model. Chicago: University of. Chicago Press, 1990.
[60]
"Playhouse 90" In Lonely Expectation (TV Episode 1959) - IMDb
In Lonely Expectation ... A woman converts a house into a home for unwed mothers and tries to help a teenage girl decide whether to keep her child or agree to an ...
[61]
CTVA US Anthology - "Playhouse 90" (CBS) Season 3 (1958-59)
3.26 [105] Playhouse 90: IN LONELY EXPECTATION 02Apr1959 CBS Thur Written by Mayo Simon Directed by Franklin Schaffner cast. Diane Baker .......... Sheila ...
[62]
The Golden Age of Television
This exhibit explores what many have labeled the “Golden Age of Television”: the live anthology dramas of the 1950s and early 1960s.
[63]
About Us - Expectation TV
Formed in 2017 by Tim Hincks and Peter Fincham, Expectation has been named one of the ten most influential Indies in the U.K. by Broadcast magazine.Missing: history | Show results with:history
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BBC Worldwide invests in Expectation Entertainment - Media Centre
Feb 16, 2017 · Expectation is a newly created independent production company. Its founders, Peter Fincham and Tim Hincks, bring extremely strong creative ...
[65]
The Grand Tour - Wikipedia
Expectation Entertainment (2019–2023); Grand Tour Productions Ltd (2019–2023) ... Series 4–6 (2019–2024): Roadtrip specials. edit. On 13 December 2018 ...
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The Grand Tour: Expectation to take production team under wing
Sep 30, 2019 · Expectation will house The Grand Tour as well as other projects from presenters Jeremy Clarkson, Richard Hammond, and James May.
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I Literally Just Told You (TV Series 2021– ) - IMDb
Rating 7.2/10 (288) I Literally Just Told You: Created by Richard Bacon. With Jimmy Carr, Maggie ... Channel 4 Television · Expectation Entertainment · See more company ...
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'I Literally Just Told You': C4 Recommissions Richard Bacon Format
Mar 9, 2022 · Produced by The Big Narstie Show indie Expectation, I Literally Just Told You from British formats creator Bacon sees four contestants given the ...
[69]
The Big Narstie Show - Wikipedia
The Big Narstie Show was produced for Channel 4 by Expectation and Dice Productions Entertainment.Production · Episodes · Series 1 (2018) · Series 4 (2021)
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The Big Narstie Show - Expectation TV
The Big Narstie Show is an anarchic Channel 4 studio chat show with celebrity guests, music, sketches, and co-hosted by Mo Gilligan. It was a huge hit.
[71]
Channel 4 recommissions The Big Narstie Show
Aug 22, 2018 · The Big Narstie Show was one of Channel 4's most popular programmes for young audiences. It was up 94% on share of 16-24 years old viewers, up ...
[72]
Ozzy Osbourne: Coming Home - Wikipedia
Sharon & Ozzy Osbourne: Coming Home is a documentary film produced by Expectation TV for BBC One. The documentary features footage of the Osbourne family ...Missing: 2002 | Show results with:2002
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Ozzy Osbourne Documentary 'Home To Roost' Could Still Air On BBC
Jul 23, 2025 · Three years on from the BBC's greenlight, however, Home to Roost has yet to premiere, despite producer Expectation having filmed with the family ...