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Sacara umum '''ekspektasi''' nyaeta tetempoan nu leuwih mungkin ngeunaan kajadian. Hasil nu kurang nguntungkeun ngakibatkeun naekna [[emotion]] '''kateupanujuan'''. Lamun sababaraha kajadian ngarupakeun hal nu teu sakabehna diperkirakeun disebutna [[surprise]]. Tempo oge [[anticipation]].
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Dina [[kamungkinan]] (hususna dina [[gambling]]), '''nilai ekspektasi''' (atawa '''ekspektasi''') tina variabel random ngarupakeun jumlah probabiliti unggal hasil nu mungkin tina sababaraha percobaan ku hasilna ("nilai"). Mangka, ieu gmabarangambaran rata-rata ngeunaan hiji "ekpektasi" keur meunang unggal tarohan lamun eta tarohan identik teu sarua unggal waktu ''pengulangan''. Catetan, nilai eta sorangan teu bisa di-ekspektasi sacara umum, saperti teu mirip atawa kajadian nu teu mungkin.
Contona, [[Roulette]] Amerika ngabogaan 38 hasil kamungkinan. Tarohan disimpen dina hiji angka bayaran 35-ka-1 (ieu hartina yen manehna mayar 35 kali tarohan, whilesabalikna alsooge hisalungan betmanehna is returneddibalikeun, togetherbareng he getsjeung 36 timeskali hisdina betalunganna). SoMangka thenilai expectedekspektasi valuehasil ofkauntungan thetina profit resulting from aunggal $1 betalungan ondina ahiji singlewilangan number isnyaeta, considering alltempo 38 possiblesakabeh hasil nu outcomesmungkin: ( -1 × 37/38 ) + ( 35 × 1/38 ), whichieu is aboutkira-kira -0.0526. ThereforeSanajan onehiji expectsekspektasi, ondina average, toleungit loseleuwih overti 5 cents forkeur everyunggal dollar betalungan.
InSacara generalumum, iflamun ''X'' is angarupakeun [[variabel random variable]] defineddihartikeun on adina [[probability space|rohangan probabiliti]] (Ω, ''P''), then themangka '''expectednilai valueekspektasi''' E''X'' oftina ''X'' is defineddirumuskeun assalaku
:<math>\operatorname{E}X = \int_\Omega X dP</math>
wherenumana thengagunakeun [[Lebesgue integration|Lebesgue integral Lebesgue]]. isCatetan employed.yen Noteteu thatsakabeh not allvariabel random variablesngabooan havenilai an expected valueekspektasi, sincelamun theintegralna integral may notteu existaya. TwoDua variables with the samevariabel [[probability distribution|sebaran probabiliti]] willnu sarua bakal havengabogaan thenilai sameekspektasi expectednu valuesarua.
IfLamun ''X'' is anyaeta [[discrete random variable|variabel random diskrit]] with valuesmibanda nilai''x''<sub>1</sub>, ''x''<sub>2</sub>, ... andsarta correspondingprobabiliti probabilitiespakait ''p''<sub>1</sub>, ''p''<sub>2</sub>, ... whichnu addditambahkeun up toka 1, thenmangka E''X'' canbisa beiitung computedsalaku asjumlah the sum oratawa [[infinite series|seriesderet]]
:<math>\operatorname{E}X = \sum_i p_i x_i</math>
assaperti indina theconto ''gambling'' exampledi mentioned aboveluhur.
If theLamun [[probability distribution|sebaran probabiliti]] of ''X'' admitsaya adina [[probability density function|fungsi probabiliti densiti]] ''f''(''x''), then themangka expectednilai valueekspektasi canbisa bediitung computedku as
:<math>\operatorname{E}X = \int_{-\infty}^\infty x f(x) dx.</math>
TheOperator expectednilai value operatorekspektasi (oratawa '''expectationoperator operatorekspektasi''') E isngarupakeun [[linear operator|linearlinier]] in thedi sensehal thatieu
:E(''aX'' + ''bY'') = ''a'' E''X'' + ''b'' E''Y''
forkeur anyunggal twodua randomvariabel variablesrandom ''X'' andjeung ''Y'' (whichnu needperlu todihartikeun bedina definedrohangan onprobabiliti thenu same probability spacesarua) andsarta any twodua [[real number|wilangan riil]]s ''a'' andjeung ''b''.
TheNilai expectedekspektasi values of the powers of''power'' ''X'' are called thedisebut ''moments'' of ''X''; the [[moment about the mean|moments about the mean]] of ''X'' areoge alsodihartikeun definedsalaku asnilai certainekspektasi expectednu valuespenting.
Umumna, operator nilai ekspektasi teu multiplicative, contona E(''XY'') teu sarua jeung E''X'' E''Y'', iwal ti lamun ''X'' jeung ''Y'' variabel [[statistical independence|bebas]]. Bedana, sacara umum, ningkat jadi [[kovarian]] jeung [[correlation|korelasi]].
Untuk estimasi nilai ekspektasi variabel random, bisa dipake nilai ukuran ''pengulangan'' variabel sarta ''perhitungan'' hasil tina [[arithmetic mean]]. Estimasi ieu nilai ekspektasi nu sabenerna sarta sipat nga-''minimal''-keun kuadrat kasalahan nilai nilai ekspektasi.
To empirically estimate the expected value of a random variable, one repeatedly measures values of the variable and computes the [[arithmetic mean]] of the results. This estimates the true expected value and has the property of minimizing the sum of the squares of the errors away from the expected value.
[[de:Erwartungswert]]
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