izzako/sundanese-pixelgpt huggingface dataset:
Lamun "X" nyaéta variabel random diskrit mibanda nilai"x"1, "x"2, ... sarta probabiliti pakait "p"1, "p"2, ... nu ditambahkeun ka 1, mangka E"X" bisa iitung salaku jumlah atawa deret formula_2 saperti dina conto "gambling" di luhur
Lamun sebaran probabiliti "X" aya dina fungsi probabiliti densiti "f"("x"), mangka nilai ekspektasi bisa diitung ku formula_3 Operator nilai ekspektasi (atawa operator ekspektasi) E mangrupa linier di hal ieu E("aX" + "bY") = "a" E"X" + "b" E"Y" keur unggal dua variabel random "X" jeung "Y" (nu perlu dihartikeun dina rohangan probabiliti nu sarua) sarta dua wilangan riil "a" jeung "b"
Nilai ekspektasi "power" "X" disebut "moments" "X"; moments about the mean "X" ogé dihartikeun salaku nilai ekspektasi nu penting
Umumna, operator nilai ekspektasi teu multiplicative, contona E("XY") teu sarua jeung E"X" E"Y", iwal ti lamun "X" jeung "Y" variabel bebas
Bedana, sacara umum, ningkat jadi kovarian jeung korelasi
keur estimasi nilai ekspektasi variabel random, bisa dipaké 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
Upamana aya susunan data, X={x1, x2, ..., xn} jeung hubungan beuratna , W={w1, w2, ..., wn} weighted méan diitung maké rumus formula_1 Catetan yén lamun kabéh beuratna sarua, weighted méan sarua jeung arithmetic mean
Balik ka summary statistics -- central tendency
Median ngagambarkeun harti teknis "middle" (tengah) dumasar kana "perasaan"
Median dina Statistik
Dina statistik, kecap median mangrupa nilai anu ngabagi sampel satengah ka luhur jeung satengah ka handap
Keur manggihkeun "median", susun data observasi ti nilai panghandapna nepi ka pangluhurna terus pilih salah sahiji anu di tengah
If there are an even number of observations, take the mean of the two middle values
When we use the "median" to describe what the observations have in common, there are several choices for a méasure of variability, the range, the interquartile range, sarta simpangan mutlak
Since the median is the same as the "second quartile", its calculation is illustrated in the article on quartiles
The median is primarily used for skewed distributions, which it represents more accurately than the arithmetic mean
Consider the set {1, 2, 2, 2, 3, 9}
The median is 2 in this case, as is the mode, and it might be seen as a better indication of central tendency than the arithmetic mean of 3.166
The median is also the central point which minimises the average of the absolute deviations; in the example above this would be (1+0+0+0+1+7)/6=1.5 using the median, while it would be 1.944... using the méan
Even though sorting "n" items takes in general O("n" log "n") operations, by using a recursive "Divide-and-Conquer" algorithm the median of "n" items can be computed with only O("n") operations
Calculation of medians is a popular technique in summary statistics and summarizing statistical data, since it is simple to understand and éasy to calculate, while also giving a méasure that is more robust in the presence of outlier values than is the mean
The difference between the median and the méan is less than or equal to one standard deviation
Median of a Triangle
In a triangle, a median is a line joining a vertex to the midpoint of the opposite side
It divides the triangle into two parts of equal area
The three medians intersect in the triangle's centroid or center of mass, and two-thirds of the length of éach median is between the vertex and the centroid, while one-third is between the centroid and the midpoint of the opposite side
Any other lines which divide the aréa of the triangle into two equal parts do not pass through the centroid
Median strip of a road
On an expressway, motorway, or autobahn, the median is the strip of grass or the wall which separates opposing lanes of traffic
This is necessary because of safety concerns, due to the high speed of automobiles on both sides, and the potential danger of a disastrous héad-on collision at the combined speed of both vehicles
Medians function secondarily as "green areas", béautifying roadways
Some jurisdictions mow their medians, others scatter wildflower seeds which germinate and re-seed themselves every year, while still others créate extensive plantings of trees, shrubs, herbaceous perennials and decorative grasses
Where space is at a premium, dense hedges of shrubs filter the héadlights of oncoming traffic and provide a resilient barrier
Variabel acak bisa disebut hasil operasi analisa numerik non-deterministik atawa "nga-bentuk" percobaan non deterministik keur ngahasilkeun hasil acak
Conto, ngagorolongkeun duit receh jeung ngarekam hasilna dina variabel random nu hasilna { 1, 2, 3, 4, 5, 6 }
Milih jalma sacara acak jeung ukur jangkungna mangrupa conto séjén tina variabel acak
Sacara matematik, variabel acak dihartikeun hiji fungsi ukuran tina rohangan probabiliti keur ukuran rohangan
Ukuran rohangan nyaéta nilai variabel ruang nu mungkin, umumna dicokot keur dijadikeun wilangan riil nu mibanda Borel σ-algebra, sarta bakal salawasna dipaké dina ieu ensiklopedia, iwal tina dina hal husus
Lamun variabel acak "X":Ω-&gt;R aya dina ruang probabiliti (Ω, "P") dibérékeun, urang bisa nanyakeun saperti kieu "Sabaraha mungkin nilai "X" leuwih gedé tinimbang 2?"
Ieu sarua jeung kamungkinan kajadian {"s" dina Ω : "X"("s") &gt; 2} nu salawasna ditulis P("X" &gt; 2) keur nyingketna
Sakabéh rekaman rentang hasil kamungkinan tina nilai-réal variabel random "X" bakal ngahasilkeun sebaran probabilitas "X"
Sebaran kamungkinan "poho" ngeunaan bagéan ruang probabiliti dipaké keur ngartikeun "X" jeung ngan direkam dina variasi nilai "X"
Saperti sebaran kamngkinan salawasna kawengku dina fungsi kumulatif sebaran formula_1 sarta kadangkala maké ogé fungsi probabilitas densitas
Dina watesan teori ukuran, dipaké variabel acak "X" keur "dorong-hareup" ukuran "P" dina Ω kana ngukur d"F" dina R. Dina kaayaan ruang probabiliti Ω mangrupa alat tenis dipaké keur ngajamin ayana variabel acak sarta kadang-kadang keur nyusunna
Dina kaperluan praktis, leuwih ilahar dina sakabéh ruang Ω sarta nyimpen hiji ukuran dina R nu nangtukeun ukuran 1 ka sakabéh garis rill, contona dipaké dina sebaran probabiliti keur gaganti variabel acak
Fungsi variabel acak
Lamun urang ngabogaan variabel random "X" on Ω jeung hiji fungsi ukuran "f":R-&gt;R, maka "Y"="f"("X") ogé jadi variabel random dina Ω, salila fungsi komposisi ukuran bisa diukur
Sababaraha prosedur ngijinkeun keur ngarobah tina ruang probabiliti (Ω,P) kana (R,dF"X") bisa dipaké keur nangtukeun sebaran probabiliti "Y"
Fungsi sebaran kumulatif "Y" nyaéta formula_2 Conto
Anggap "X" nilai-riil variabel acak sarta anggap "Y" = "X"2
Mangka, formula_3 Lamun "y" &lt; 0, mangka P("X"2 ≤ "y") = 0, mangka formula_4 Lamun "y" ≥ 0, mangka formula_5 mangka formula_6 Momen
Sebaran probabiliti variabel random salawasna dicirikeun ku jumlah paramater nu saeutik, ogé mibanda "interpretasi" praktis, contona cukup nyaho "nilai average"
Hal ieu kawengku ku konsép matematik nilai ekspektasi variabel random, dilambangkeun ku E["X"]
Catetan yén ilaharna , E["f"("X")] teu sarua jeung "f"(E["X"])
Lamun "nilai average" dipikanyaho, mangka sabaraha jauh tipikal nilai "X" tina nilai average bisa dijawab ku varian sarta simpangan baku variabel random
Sacara matematik, ieu disebutna (generalisasi) masalah momen: keur kelas variabel random "X" nu dibéré, téangan fungsi kumpulan {"fi"} saperti nilai ekspektasi E["fi"("X")] nu sakabéhna nyirikeun distribusi variabel random "X"
Kasaruaan variabel acak
Aya sababaraha béda harti dina variabel acak nu bisa dianggap sarua
Dua variabel acak bisa sarua, yakin sarua, sarua dina méan atawa sarua dina sebaran
Dina raraga leuwih ngajéntrékeun, di handap ieu sababaraha hal ngeunaan kasaruaan dijéntrékeun
Sarua dina sebaran
Dua variabel acak "X" jeung "Y" "sarua dina sebaran" lamun formula_7 Bakal sarua dina sebaran, variabel acak teu perlu kapanggih dina rohang probabilitas nu sarua, tapi teu kaleungitan kailaharan nu bisa dijieun kana variabel acak dina rohang probabilitas nu sarua
Lambang nu sarua di sebaran pakait jeung lambang jarak saterusna antara sebaran probabilitas, formula_8 nu mana dumasar kana uji Kolmogorov-Smirnov
Sarua dina mean
Dua variabel acak "X" jeung "Y" "sarua dina mean ka-p" lamun momen ka-"p" |"X" − "Y"| sarua jeung enol, nyaéta, formula_9 Kasaruaan dina méan ka-"p" nyababkeun kasaruan dina méan ka-"q" keur sakabéh "q"&lt;"p"
Saperti dina kasus saméméhna, hal ieu jarak pakait antara variabel acak, dingaranan formula_10 Ampir pasti sarua
Dua variabel acak "X" jeung "Y" "ampir pasti sarua" lamun, ngan jeung lamun, probabilitasna béda ti enol: formula_11 Keur sakabéh kaperluan praktis dina téori probabilitas, notasi kasaruaan ieu sarua kuatna jeung sarua dina sabenerna
Hal ieu pakait jeung jarakna: formula_12 nu mana 'sup' dina kasus ieu ngagambarkeun essential supremum dina pamadegan teori ukuran
Tungtungna, dua variabel acak "X" jeung "Y" disebut "sarua" lamun maranéhna sarua salaku fungsi dina rohang probabilitas, nyaéta, formula_13 Konvergen
Loba statistik matematik nu kabuktian konvergen keur sekuen nu ditempo dina variabel acak; tempo hukum wilangan loba jeung teorema limit tengah
Aya sababaraha itungan nu mana unggal hiji ("X""n") variabel acak bisa konvergen kana variabel acak "X"
Ngeunaan hal ini dijéntrékeun dina artikel Konvergen variabel acak
Di handap ieu mangrupa conto tina integer random i, 1 ≤ i ≤ 100: 17 12 17 89 64 4 62 6 82 80 61 100 19 7 35 4 23 43 49 69 4 81 64 52 33 59 56 56 46 25 2 44 34 73 58 48 94 18 65 47 73 16 69 26 26 65 35 65 64 2
26 65 35 65 64 2 59 36 52 77 52 14 79 42 71 82 60 28 72 96 77 72 78 58 71 44 99 41 41 80 53 67 7 66 49 86 94 85 47 27 1 6 86 50 32 26 60 79 94 53 72 98 78 46 73 50 49 3 77 57 56 23 20 70
77 57 56 23 20 70 1 58 42 72 16 84 96 44 42 76 19 71 57 17 34 66 68 63 100 37 38 68 52 52 42 86 15 53 76 59 43 94 67 21 74 73 85 16 12 45 57 7 4 22 23 74 15 63 80 65 76 88 39 39 100 96 85 64
39 39 100 96 85 64 16 55 62 50 71 27 48 95 96 30 65 33 71 50 39 1 70 99 55 74 2 74 98 48 99 90 28 66 41 17 80 35 8 30 85 41 68 18 46 86 91 40 20 43 71 95 48 40 79 88 77 49 81 52 15 8 11 51
81 52 15 8 11 51 26 99 8 28 37 47 37 17 30 27 39 33 65 8 31 73 48 96 41 78 9 89 72 16 61 48 73 90 39 34 7 41 1 87 48 83 41 64 61 47 71 2 35 66 74 29 74 7 61 22 46 46 4 59 23 79 33 7
4 59 23 79 33 7 31 41 54 63 91 81 58 66 83 24 37 84 16 55 9 52 92 69 44 27 57 38 70 37 33 23 24 18 74 20 87 73 28 85 34 31 76 25 6 38 15 73 16 79 83 94 21 52 34 19 66 5 97 33 100 63 36 100
97 33 100 63 36 100 4 63 84 8 21 21 92 60 72 22 25 80 23 8 10 10 63 44 14 86 47 17 45 4 18 21 44 27 88 10 92 90 27 54 73 68 13 15 68 31 4 83 46 97 97 32 12 66 66 87 100 75 99 75 73 16 86 90
99 75 73 16 86 90 66 51 59 80 87 40 35 21 76 65 74 73 26 41 17 67 88 54 42 62 98 78 19 29 60 79 19 76 13 95 68 76 86 47 91 23 25 50 57 27 97 30 16 82 5 7 31 72 64 18 32 100 54 18 51 66 38 74
54 18 51 66 38 74 91 75 41 81 21 32 96 78 90 9 82 21 84 80 65 72 52 17 81 50 1 90 14 45 11 76 91 31 20 93 30 30 66 10 20 37 89 3 71 35 96 82 11 4
"Artikel ieu ngeunaan matematik
Tempo ogé variance (land use)."
Dina tiori probabiliti sarta statistik, varian tina variabel acak mangrupa ukuran tina statistical dispersion, nu nembongkeun sabaraha jauh tina nilai ekspektasi nu dijelaskeun di dieu
Varian nilai-real variabel acak mangrupa momen mean nu kadua, ogé cumulant nu kadua (cumulants béda jeung central moments ngan dina tingkat 4 atawa saluhureunna)
Lamun μ = E("X") mangrupa nilai ekspektasi tina variabel acak "X", mangka varian nyaéta formula_1 contona, varian mangrupa nilai ekspektasi kuadrat simpangan "X" tina méan-na sorangan
Jadi varian mangrupa "simpangan mean kuadrat"
Varian variabel random"X" dituliskeun minangka var("X")
Catetan loba sebaran, saperti sebaran Cauchy, teu ngabogaan varian sabab nyimpang tina integral
Dina hal séjén, lamun sebaran teu ngabogaan nilai ekspektasi, mangka teu ngabogaan varian oge
Hal nu teu bener: sebaran ngabogaan nilai ekspektasi tapi teu ngabogaan varian
Lamun varian dihartikeun, bisa disimpulkeun yén varian teu pernah negatip sabab kuadrat bakal positip atawa nol
Lamun métodeu keur ngitung varian hasilna negatip, geus tangtu aya kasalahan, ilaharna dina nangtukeun algoritma
Satuan varian nyaéta kuadrat tina satuan sebaran
Mangka, varian tina susunan ukuran jangkung dina sentiméter nyaéta sentiméter kuadrat
Kanyataan ieu kurang méré nah sarta statistikawan leuwih ilahar ngagunakeun akar varian, simpangan baku sarta ngagunakeun ieu nilai minangka kasimpulan dispersi
Ieu bisa dibuktikeun sacara gampang tina harti yén varian moal gumantung kana nilai méan formula_2
Dina hal ieu, lamun variabel "disimpen" antara "b" jadi "X"+"b", varian hasil variabel random beulah kénca teu kacekel
Sacara jelas, lamun variabel dikalikeun ku faktor skala "a", varian mangrupa hasil kali nyaéta "a2"
Sacara resmi, lamun "a" jeung "b" mangrupa konstanta riil sarta "X" mangrupa variabel acak mangka varian dihartikeun ku, formula_3 Formula séjén keur varian saperti dina garis lurus nu dumasar kana harti di luhur nyaéta: formula_4 Ieu mangrupa rumus nu geus ilahar dipaké keur ngitung varian dina kaperluan praktis
Salah sahiji alesan varian leuwih sering dipaké tinimbang ukuran dispersi séjénna nyaéta yén varian jumlah variabel acak bébas sarua jeung jumlah varian-na
(Kaayaan nu leuwih lemah tinimbang bébas, disebutna "uncorrelatedness" atawa taya hubungan)
Sacara umum, formula_5 Di dieu formula_6 nyaéta kovarian, sarua jeung nol keur variabel nu taya hubungan
Populasi varian jeung sampel varian
Dina statistik, konsép varian ogé digunakeun keur ngajelaskeun susunan data
Waktu susunan data mangrupa populasi, mangka disebut "populasi varian"
Waktu susunan data mangrupa sample, mangka disebutna "sampel varian"
Populasi varian tina populasi "yi" di mana "i = 1, 2, ..., N" dirumuskeun ku formula_7 di mana formula_2 mangrupa populasi méan
Dina prakték, waktu kaayaan populasi gedé, geus ilahar yén teu mungkin manggihkeun nilai populasi varian nu pasti, sabab kawengku ku waktu, béaya jeung sumber séjénna
métodeu nu geus ilahar dipaké keur estimasi populasi varian nyaéta sampling
Waktu estimasi populasi varian ngagunakeun "n" random samples "xi" di mana "i = 1, 2, ..., n", nuturkeun rumus di handap ieu mangrupa unbiased estimator: formula_9 di mana formula_10 mangrupa sampel méan
Catetan yén "n-1" dina pembagi di luhur jelas béda jeung persamaan keur ngitung populasi varian
Sumber nu geus ilahar ngabingungkeun nyaéta watesan "sampel varian" jeung notasi "s2" bisa jadi nempo kana unbiased éstimator séjén tina populasi varian nu dirumuskeun di luhur, sarta kumaha cara mastikeun varian tina sampel, diitung ku "n" tinimbang "n-1"
Sacara rasa, ngitung varian ku ngabagi maké "n" tinimbang "n-1" méré hasil populasi varian "underestimate"
Hal ieu sabab urang ngagunakeun sampel méan formula_10 keur estimasu populasi méan formula_2, nu teu dipikanyaho
Dina prakten, keur "n" nu gedé, bédana salawasna kurang ti hiji
"Tempo ogé algoritma keur ngitung varian."
Lamun "X" mangrupa nilai-vector- variabel random, nu mibanda nilai dina"R""n", sarta dipinkanyaho minangka vektor kolom, mangka generalisasi sacara alami tina varian nyaéta E(("X" − μ)("X" − μ)′), nu mana μ = E("X") sarta "X" ′ mangrupa transpos "X", sarta jadi vektor baris
Varian ieu mangrupa nonnegative-definite matriks kuadrat, umumna dianggap minangka covariance matrix
Lamun "X" mangrupa nilai-kompleks variabel random, mangka varian nyaéta E(("X" − μ)("X" − μ)*), nu mana "X"* mangrupa complex conjugate "X"
Varian ieu mangrupa angka riil nonnegative
Watesan varian mimiti dikenalkeun ku Ronald Fisher taun 1918 dina "paper"-na "The Correlation Between Relatives on the Supposition of Mendelian Inheritance" Rujukan
Dina probabilitas jeung statistika, simpangan baku biasa digunakeun keur ngukur statistical dispersion
Simpangan baku dihartikeun ogé akar kuadrat tina varian
Hal ieu dimaksudkeun keur ngukur "dispersi" nyaéta 1) angka non-négatif; jeung 2) miboga unit nu sarua jeung datana
Bisa dibédakeun antara simpangan baku σ (sigma) tina "populasi" atawa variabel acak, jeung simpangan baku "s" tina "sampel"
Rumusna dijelaskeun di handap
Watesan simpangan baku dina statistik mimiti dikenalkeun ku Karl Pearson ("On the dissection of asymmetrical frequency curves", 1894)
Interprétasi jeung Pamakéan
Sacara gampang, simpangan baku nyebutkeun sabaraha jauh unggal anggota sampel atawa populasi tina nilai mean sampel atawa populasi
Nilai simpangan baku anu gedé nunjukkeun yén anggota anggota nu dimaksud jauh tina "mean"
Nilai simpangan baku leutik nunjukkeun yén anggota nu dimaksud raket atawa aya sabudeureun "mean"
Conto, susunan {0,5,9,14} jeung {5,6,8,9} mibanda nilai "mean" 7, tapi nilai susunan data nu kadua mibanda nilai simpangan baku anu leuwih leutik
Simpangan baku ogé biasa dipaké keur ngukur kateupastian
Conto dina elmu fisika, waktu ngalakukeun "pengulangan" measurements simpangan baku tina pengukuran nyaéta precision tina éta pengukuran
Waktu keur mutuskeun yén ukuran sarua jeung prediksi, ukuran simpangan baku mangrupa hal anu kacida pentingna: lamun ukuran kacida jauhna tina prediksi (ku jarak ukuran simpangan baku), bisa dianggap yén ukuran patojaiah jeung prediksi
Hal ieu ngajadikeun pamikiran yén rentang nilaina kaluar tina anu diperkirakeun lamun prediksina bener
Tempo prediction interval
Harti jeung cara pondok ngukur simpangan baku
Anggap nilai tina hiji populasi "x"1...,"x""N" (nu mangrupa wilangan riil)
Mean populasi dirumuskeun ku formula_1 (tempo kumpulan notasi) sarta simpangan baku populasi dirumuskeun ku formula_2 Cara gampang ngitung simpangan baku dina jumlah nu sarua dirumuskeun ku formula_3 Simpangan baku variabel acak "X" dihartikeun ku formula_4 Catetan yén teu sakabéh variabel random mibanda simpangan baku, lamun nilai ekspektasi euweuh
Lamun variabel random "X" dicokot tina nilai "x"1...,"x""N" ku probabiliti nu sarua, simpangan baku bisa diitung maké rumus saméméhna
Dina kaayan nilai sampel "x"1...,"x""n" ti populasi nu gedé, sababaraha pangarang ngartikeun "sampel simpangan baku" ku formula_5 Alesan keur harti ieu yén "s"2 mangrupa unbiased estimator keur varian σ2 ti populasi
Catetan "s" sorangan "lain" unbiased éstimator keur simpangan baku σ; hal ieu cenderung underestimate ti populasi simpangan baku
Aturan keur data sebaran normal
Dina praktékna, biasa diasumsikeun yén data ngabogaaan sebaran normal
Lamun asumsi ieu bisa diyakinkeun, mangka nilai 68% dina 1 simpangan baku jauh tina méan, nilai 95% dina dua simpangan baku jauh tina méan, sarta nilai 99.7% nutupan di jero 3 simpangan baku tina méan
Ieu dikanyahokeun salaku "aturan 68-95-99.7"
Hubungan simpangan baku jeung "mean"
Méan jeung simpangan baku tina susunan data mangrupa hal nu raket sarta umumna ditulis babarengan
Hal nu penting, simpangan baku nyaéta ukuran "alami" dispersi statistik lamun pusat data diukur ku méan
Rumus pastina nyaéta: suppose "x"1...,"x""N" are réal numbers and define the function formula_6 Ngagunakeun kalkulus, teu hésé keur nembongkeun yén σ("r") mibanda "unique minimum" keur formula_7 Interpretasi geometrik
Keur ngarti géometri leuwih jéntré, urang mimitian ku populasi tina tilu nilai, "x"1,"x"2,"x"3
Hartina yén titik "P"= ("x"1,"x"2,"x"3) aya dina R3
Anggap garis "L" = {("r","r","r") : "r" aya dina R}
Ieu mangrupa "diagonal utama" nu ngaliwatan aslina
Lamun tilu nilai tadi sarua, mangka simpangan baku sarua jeung nol sarta "P" bakal nutupan "L"
Mangka taya alesan ke nganggap simpangan baku pakait jeung "jarak" "P" ka "L"
Ieu mangrupa kasus nu bener
Pindah ortogonalitas ti "P" kana garis "L", salah sahiji titik sasaran formula_8 nu mana kordinat nilai méan dimimitian
Aljabar sederhana nunjukkeun yén jarak antara "P" jeung "R" (hartina sarua jeung jarak antara "P" jeung garis "L") dirumuskeun ku σ√"3"
Rumus analogna (3 digantikeun ku "N") ogé "valid" keur populasi nilai "N" values; mangka saterusna dinaR"N"
Simpangan baku salaku tingkat kapercayaan
Dina percobaan ilmiah, hiji kapercayaan tina ukuran kajadian mangrupa hasil tina signal tinimbang ngan sakadar tina ramalan statistik
Mangka luhurna tingkat kapercayaan, mangrupa ukuran kajadian tinimbang ramalan
Ukuran dispersi statistik kudu ngahasilkeun angka nol lamun sakabéh data identik, sarta kudu naék lamun datana leuwih béda
Ukuran dispersi pangpentingna nyaéta simpangan baku, akar kuadrat varian
Ukuran séjénna nyaéta rentang, interquartile range, sarta rata-rata simpangan mutlak, sarta dina kasus kategori variabel random, entropi diskrit
Tara pernah negatip; nilai pangleutikna nu mungkin nyaéta nol
Ukuran dispersi statistik sabagian dipaké lamun lokasi invariant, sarta liniér dina skala
Maka lamun variabel random "X" mibanda dispersi "SX" mangka transpormasi liniér "Y" = "aX" + "b" keur wilangan riil "a" jeung "b" kudu mibanda dispersi "SY" = "aSX"
Tempo ogé kasimpulan statistik
Karl Péarson (27 Maret 1857 – 27 April 1936) loba kontribusina dina pengembangan statistics saperti disiplin elmu nu sarius
Anjeunna ngadegkeun Jurusan Applied Statistics di University College London taun 1911; mangrupa universitas mimiti nu mibanda jurusan statistik di dunya
Karl Péarson lahir di London kaping 27 Maret, 1857
Anjeunna diatik sacara privat di University College School sarta dituluykeun ka King's College, Cambridge pikeun diajar Matématika
He then spent part of 1879 and 1880 studying medieval and 16th-century German literature at the universities of Berlin and Heidelberg – in fact, he became sufficiently knowledgéable in this field that he was offered a post in the German department at Cambridge University
His next career move was to Lincoln's Inn, where he réad law until 1881 (although he never practised)
After this, he returned to mathematics, deputising for the mathematics professor at King's College London in 1881 and for the professor at University College London in 1883
In 1884, he was appointed to the Goldshmid Chair of Applied Mathematics and Mechanics at University College London
1891 saw him also appointed to the professorship of Geometry at Gresham College; here he met W.F.R
Weldon, a zoologist who had some interesting problems requiring quantitative solutions
The collaboration, in biometry and evolutionary théory, was a fruitful one and lasted until Weldon died in 1906
Weldon introduced Péarson to Francis Galton, who was interested in aspects of evolution such as heredity and eugenics
Galton died in 1911 and left the residue of his estate to the University of London for a Chair in Eugenics
Péarson was the first holder of this chair, in accordance with Galton's wishes
He formed the Department of Applied Statistics (with financial support from the Drapers' Company), into which he incorporated the Biometric and Galton laboratories
He remained with the department until his retirement in 1933, and continued to work until his déath in 1936
Péarson married Maria Sharpe in 1890, and between them they had 2 daughters and a son
The son, Egon Sharpe Pearson, succeeded him as héad of the Applied Statistics Department at University College
Aside from his professional life, Péarson was active as a prominent freethinker and socialist
He gave lectures on such issues as "the woman's question" (this was the éra of the suffragette movement in the UK) and upon Karl Marx
His commitment to socialism and its idéals led him to refuse the offer of being créated an OBE (Officer of the Order of the British Empire) in 1920, and also to refuse a Knighthood in 1935
Péarson's views on eugenics, however, would be considered deeply racist today
According to a BBC report on the history of genetics, "Pearson was a fanatic – a cold, calculating measurer of man who claimed to be a socialist, but loathed the working class."
Péarson openly advocated "war" against "inferior races," and saw this as a logical implication of his scientific work on human méasurement: "My view – and I think it may be called the scientific view of a nation," he wrote, "– is that of an organized whole, kept up to a high pitch of internal efficiency by insuring that its numbers are substantially recruited from
its numbers are substantially recruited from the better stocks, and kept up to a high pitch of external efficiency by contest, chiefly by way of war with inferior races."
Awards from professional bodies
Péarson achieved widespréad recognition across a range of disciplines and his membership of, and awards from, various professional bodies reflects this: He was also elected an Honorary Fellow of King's College Cambridge, the Royal Society of Edinburgh, University College London and the Royal Society of Medicine, and a Member of the Actuaries' Club
Contributions to statistics
Péarson's work was all-embracing in the wide application and development of mathematical statistics, and encompassed the fields of biology, epidemiology, anthropometry, medicine and social history
In 1901, with Weldon and Galton, he founded the journal "Biometrika" whose object was the development of statistical théory
He edited this journal till his déath
He also founded the journal "Annals of Eugenics" (now "Annals of Human Genetics") in 1925
He published the "Drapers' Company Research Memoirs" largely to provide a record of the output of the Department of Applied Statistics not published elsewhere
Péarson's thinking underpins many of the `classical' statistical methods which are in common use today
Some of his main contributions are: Bacaan saterusna
Most of the biographical information above is taken from "A list of the papers and correspondence of Karl Pearson (1857-1936)" held in the Manuscripts Room, University College London Library, compiled by M.Merrington, B.Blundell, S.Burrough, J.Golden and J.Hogarth and published by the Publications Office, University College London, 1983
Further references which may be of use are:
akar kuadrat Dina matematik, wilangan riil "non-négatip" "x" dilambangkeun ku formula_1 sarta ngagambarkeun wilangan riil non-négatip nu mangrupa "kuadrat" (hasil kali tina wilangan éta sorangan) nyaéta "x"
Contona, formula_2 saprak formula_3
Conto ieu nembongkeun yén akar kuadrat bisa dipaké keur ngaréngsékeun "persamaan kuadrat" saperti formula_4 atawa leuwih ilahar formula_5
Ngalegaan tina konsép akar kuadrat keur wilangan riil négatip nyaéta dina "wilangan imajinér" jeung "wilangan kompléks."
Akar kuadrat mindeng mangrupa "wilangan irasional", mikabutuh anu teuaya watesna, dijerona aya sakantetan "digit" anu henteu sarua ngawakilan désimal
Contona, formula_6 henteu bisa ditulis kalawan panjeg dina wujud désimal anu aya tungtungna atawa tuluy-tuluyan
Sarua jeung, éta teu bisa diwakilan ku hiji "pecahan"/fraction anu mana eta "pembilang" jeung "penyebutnya" téh ngaruntuy/integer
Sanajan kitu, sanajan ieu sarua pisan panjangna tina diagonal tina hiji pasagi tina hiji sisi panjang
Pamanggihna yén formula_6 ieu henteu rasional di tembrakeun ku Pythagoreans
Lambang akar kuadrat (√) munggaran dipaké dina abad ka-16
Kawasna asalna tina bentuk singget pikeun r, tina Basa Latin "radix" (hartina "akar")
The following important properties of the square root functions are valid for all positive réal numbers "x" and "y": formula_8 formula_9 formula_10 for every real number "x" (see absolute value) formula_11 Fungsi akar kuadrat umumna metakeun wilangan rasional ka wilangan aljabar; √"x" is rational if and only if "x" is a rational number which, after cancelling, is a fraction of two perfect squares
In particular, √2 is irrational
In geometrical terms, the square root function maps the area of a square to its side length
Suppose that "x" and "a" are réals, and that "x"2="a", and we want to find "x"
A common mistake is to "take the square root" and deduce that "x" = √"a"
This is incorrect, because the square root of "x"2 is not "x", but the absolute value |"x"|, one of our above rules
Thus, all we can conclude is that |"x"| = √"a", or equivalently "x" = ±√"a"
In calculus, for instance when proving that the square root function is continuous or differentiable or when computing certain limits, the following identity often comes handy: formula_12 It is valid for all non-negative numbers "x" and "y" which are not both zero
The function "f"("x") = √"x" has the following graph, made up of half a parabola lying on its side: The function is continuous for all non-negative "x", and differentiable for all positive "x" (it is not differentiable for "x"=0 since the slope of the tangent there is ∞)
Its derivative is given by formula_13 Its Taylor series about "x" = 1 can be found using the binomial theorem: formula_14 formula_15 for |"x"| &lt; 1
Computing square roots
Pocket calculators typically implement good routines to compute the exponential function and the natural logarithm, and then compute the square root of "x" using the identity formula_16 The same identity is exploited when computing square roots with logarithm tables or slide rules
A commonly used algorithm for approximating √"x" is known as the "Babylonian method" and is based on Newton's method
It proceeds as follows: This is a quadratically convergent algorithm, which méans that the number of correct digits of "r" roughly doubles with éach step
This algorithm works equally well in the p-adic numbers, but cannot be used to identify réal square roots with p-adic square roots; it is éasy, for example, to construct a sequence of rational numbers by this method which converges to +3 in the réals, but to -3 in the 2-adics
An exact "long-division like" algorithm
This method, while much slower than the Babylonian method, has the advantage that it is exact: if the given number has a square root whose decimal representation terminates, then the algorithm terminates and produces the correct square root after finitely many steps
It can thus be used to check whether a given integer is a square number
Write the number in decimal and divide it into pairs of digits starting from the decimal point
The numbers are laid out similar to the long division algorithm and the final square root will appéar above the original number
For éach iteration: Example: What is the square root of 152.2756
____1__2._3__4_ | 01 52.27 56 1 x 01 1*1=1 1 ____ __ 00 52 22 2x 00 44 22*2=44 2 _______ ___ 08 27 243 24x 07 29 243*3=729 3 _______ ____ 98 56 2464 246x 98 56 2464*4=9856 4 _______ 00 00 Algorithm terminates: answer is 12.34 Although demonstrated here for base 10 numbers, the procedure works for any base, including base 2
In the description above, 20 méans double the number base used, in the case of binary this would réally be 100
The algorithm is in fact much éasier to perform in base 2, as in every step only the two digits 0 and 1 have to be tested
See Shifting nth-root algorithm
Pell's equation yields a method for finding rational approximations of square roots of integers
Finding square roots in the head
Based on Pell's equation there is a methode to calculate the square root in the héad, by simply subtraction of odd numbers
Ex: Square root of 27 is: 1) 27-1 = 26 2) 26-3 = 23 3) 23-5 = 18 4) 18-7 = 11 5) 11-9 = 2 First number is 5 2 x 100 = 200 and 5 x 20 + 1 = 101 1) 200-101 = 99 Next number is 1 99 x 100 = 9900 and 51 x 20 + 1 = 1021
x 20 + 1 = 1021 1) 9900-1021 = 8879 2) 8879-1023 = 7856 3) 7856-1025 = 6831 4) 6831-1027 = 5804 5) 5804-1029 = 4775 6) 4775-1031 = 3744 7) 3744-1033 = 2711 8) 2711-1035 = 1676 9) 1676-1037 = 639 Next number is 9 The result gives us 5.19 as the square root of 27 Continued fraction methods
Quadratic irrationals, that is numbers involving square roots in the form ("a"+√b)/"c", have periodic continued fractions
This makes them éasy to calculate recursively given the period
For example, to calculate √2, we maké use of the fact that √2-1 = [0;2,2,2,2,2...], and use the recurrence relation "a""n+1"=1/(2+a"n") with "a"0=0 to obtain √2-1 to some specific precision specified through "n" levels of recurrence, and add 1 to the result to obtain √2
Square roots of complex numbers
To every non-zero complex number "z" there exist precisely two numbers "w" such that "w"2 = "z"
The usual definition of √"z" is as follows: if "z" = "r" exp("i"φ) is represented in polar coordinates with -π &lt; φ ≤ π, then we set √"z" = √"r" exp("i"φ/2)
Thus defined, the square root function is holomorphic everywhere except on the non-positive réal numbers (where it isn't even continuous)
The above Taylor series for √(1+"x") remains valid for complex numbers "x" with |"x"| &lt; 1
When the number is in rectangular form the following formula can be used: formula_22 where the sign of the imaginary part of the root is the same as the sign of the imaginary part of the original number
Note that because of the discontinuous nature of the square root function in the complex plane, the law √("zw") = √("z")√("w") is in general not true
Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that -1 = 1: formula_23 The third equality cannot be justified
(See invalid proof.)
However the law can only be wrong up to a factor -1, √("zw") = ±√("z")√("w"), is true for either ± as + or as - (but not both at the same time)
Note that √("c"2) = ±"c", therefore √("a"2"b"2) = ±"ab" and therefore √("zw") = ±√("z")√("w"), using "a" = √("z") and "b" = √("w")
Square roots of matrices and operators
If "A" is a positive definite matrix or operator, then there exists precisely one positive definite matrix or operator "B" with "B"2 = "A"; we then define √"A" = "B"
More generally, to every normal matrix or operator "A" there exist normal operators "B" such that "B"2 = "A"
In general, there are several such operators "B" for every "A" and the square root function cannot be defined for normal operators in a satisfactory manner
Positive definite operators are akin to positive réal numbers, and normal operators are akin to complex numbers
Square roots of the first 20 positive integers
√ 1 = 1 √ 2 ≈1.4142135623 7309504880 1688724209 6980785696 7187537694 8073176679 7379907324 78462 √ 3 ≈1.7320508075 6887729352 7446341505 8723669428 0525381038 0628055806 9794519330 16909 √ 4 = 2 √ 5 ≈2.2360679774 9978969640 9173668731 2762354406 1835961152 5724270897 2454105209 25638 √ 6 ≈2.4494897427 8317809819 7284074705 8913919659 4748065667 0128432692 5672509603 77457 √ 7 ≈2.6457513110 6459059050 1615753639 2604257102 5918308245 0180368334 4592010688 23230 √ 8 ≈2.8284271247 4619009760 3377448419 3961571393
√ 8 ≈2.8284271247 4619009760 3377448419 3961571393 4375075389 6146353359 4759814649 56924 √ 9 = 3 √10 ≈3.1622776601 6837933199 8893544432 7185337195 5513932521 6826857504 8527925944 38639 √11 ≈3.3166247903 5539984911 4932736670 6866839270 8854558935 3597058682 1461164846 42609 √12 ≈3.4641016151 3775458705 4892683011 7447338856 1050762076 1256111613 9589038660 33818 √13 ≈3.6055512754 6398929311 9221267470 4959462512 9657384524 6212710453 0562271669 48293 √14 ≈3.7416573867 7394138558 3748732316 5493017560 1980777872 6946303745 4673200351 56307 √15 ≈3.8729833462 0741688517 9265399782 3996108329
56307 √15 ≈3.8729833462 0741688517 9265399782 3996108329 2170529159 0826587573 7661134830 91937 √16 = 4 √17 ≈4.1231056256 1766054982 1409855974 0770251471 9922537362 0434398633 5730949543 46338 √18 ≈4.2426406871 1928514640 5066172629 0942357090 1562613084 4219530039 2139721974 35386 √19 ≈4.3588989435 4067355223 6981983859 6156591370 0392523244 4936890344 1381595573 28203 √20 ≈4.4721359549 9957939281 8347337462 5524708812 3671922305 1448541794 4908210418 51276
Informasi ngeunaan estimating paraméters bisa ditempo ogé dina;
Applikasi software (atawa disingket "app") nyaéta program komputer atawa kumpulan program nu dirancang pikeun nyadiakeun kagunaan keur end userna
Conto hususna nyaéta aplikasi program "word processor" (saperti MS Word), "spreadsheet" (saperti MS Excel), "accounting", jeung "media player"
Sababaraha aplikasi nu dibeundeul jadi hiji kadang-kadang disebut application suite
Microsoft Office, nu ngabeundeul word processor, spréadsheet katut sababaraha aplikasi liana, mangrupa salah sahiji contona
Istilah "aplikasi" bisa dipaké keur ngabédakeun tipe program ieu tina kelompok pakakas lemes utama liana nu disebut pakakas lemes sistim nu mangrupa pakakas lemes nu pakait jeung ngokolakeun tur ngagunakeun aspek-aspek sistim komputerna sorangan kawas sistim operasi, "device driver", jeung compiler
Aplikasi ogé bisa ngandung harti ngasupkeun lamaran gawé
Statistik bisnis mangrupa cabang applied statistics loba dipaké keur ngumpulkeun data salaku hasil tina bisnis atawa lembaga pamaréntah
Sipat sumber ieu pengulangan nu angger tina deret data
Hal ieu ngajadikeun deret waktu penting keur bisnis statistik
Sababaraha téhnik Marketing ogé dipaké leuwih Multivariate statistics lanjut dina segmen pasar, manajemen merk sarta téhnik posisi produk
Démografi nyaéta élmu ngeunaan dinamika populasi manusa, nu ngawengku panalungtikan ukuran, struktur, jeung sébaran populasi, sarta kumaha parobahan populasi sajalan jeung nyérélékna waktu ku ayana nu babar, maot, migrasi, jeung nambahna umur
Analisis démografis bisa nujul ka masarakat sagemblengna atawa ka golongan nu ditangtukeun ku kritéria kayaning atikan, bangsa, ageman, atawa suku
Data jeung métode
Démografi mercayakeun kana ngagunakeun data baréa-réa, kaasup sénsus kalahiran, pernikahan jeung tilar dunya
The éarliest modérn census was carried out in Britain in 1801
See also demographic statistics
In many countries, particularly in the third world, reliable demographic data are still difficult to obtain
For example, during the 1980s the population of Nigeria was widely estimated to be around 110 million, before it was established to be as little as 89 million (without adjustment for undercounting) in a census carried out in 1991
Konsép-konsép nu penting dina démografi di antarana: Note that the crude déath rate as defined above and applied to a whole population can give a misléading impression
For example, the number of déaths per 1000 péople can be higher for developed nations than in less-developed countries, despite standards of héalth being better in developed countries
This is because developed countries have relatively more older péople, who are more likely to die in a given yéar, so that the overall mortality rate can be higher even if the mortality rate at any given age is lower
A more complete picture of mortality is given by a life table which summarises mortality separately at éach age
A life table is necessary to give a good estimate of life expectancy
Among the éarliest contributions to demography were the works of Thomas Malthus
Malthus concluded that, if unchecked, populations would be subject to exponential growth
He féared that population growth would tend to outstrip growth in food production, léading to ever incréasing famine and poverty (see Malthusian catastrophe)
Later more sophisticated and réalistic modéls were presented by e.g
Gompertz and Verhulst
The demographic transition
Contrary to Malthus' predictions, natural population growth in most developed countries has diminished to close to zero, without being held in check by famine or lack of resources, as péople in developed nations have shown a tendency to have fewer children
The fall in population growth has occurred despite large rises in life expectancy in these countries
Similar trends are now becoming visible in ever more developing countries, so that far from spiralling out of control, world population growth is expected to slow markedly in the next century, coming to an eventual standstill
The change is likely to be accompanied by major shifts in the proportion of world population in particular regions
This pattern of population growth, with slow growth in preindustrial societies, followed by fast growth as the society develops and industrialises, followed by slow growth again as it becomes more affluent, is known as the demographic transition
The term demographics is often used erronéously for demography, but refers rather to selected population characteristics as used in marketing or opinion reséarch
Statistika multivariat atawa analisa multivariate statistik dina statistik ngajelaskeun kumpulan prosedur observation jeung analysis nu variabel statistikna leuwih ti hiji dina sawaktu
Aya sababaraha modél, nu gumantung kana tipe analisana nyaéta:
Téorema central limit mangrupa hasil susunan konvergen-lemah dina teori probabilitas
Ieu téorema keur ngagambarkeun yén jumlah unggal variabel acak nu kasebar bébas sacara normal (independent identically distributed atawa i.i.d) ngadeukeutan sebaran normal
Hasil ieu nerangkeun kaunikan sebaran normal
Hasil nu penting jeung kawentar éta disebutna "Teorema Central Limit Theorem"; nu museurkeun kana variabel bébas nu pasipatan sebaranna bébas tur nilai eksptasi jeung varianna "terhingga"
Several generalizations exist which do not require identical distribution but incorporate some condition which guarantees that none of the variables exert a much larger influence than the others
Two such conditions are the "Lindeberg condition" and the "Lyapunov condition"
Other generalizations even allow some "weak" dependence of the random variables
The réader may find it helpful to consider this illustration of the central limit theorem
Teorema "central limit"
Tempo "X1","X2","X3"... be a sequence of random variables which are defined on the same probability space, share the same probability distribution D and are independent
Assume that both the nilai ekspektasi μ and the simpangan baku σ of D exist and are finite
Consider the sum :"Sn"="X"1+...+"X"n. Then the expected value of "S""n" is "n"μ and its simpangan baku is σ "n"½
Furthermore, informally spéaking, the distribution of Sn approaches the normal distribution N("n"μ,σ2"n") as "n" approaches ∞
In order to clarify the word "approaches" in the last sentence, we standardize "S""n" by setting formula_1 Then the distribution of "Z""n" converges towards the standard normal distribution N(0,1) as "n" approaches ∞ (this is convergence in distribution)
This méans: if Φ("z") is the cumulative distribution function of N(0,1), then for every real number "z", we have formula_2 or, equivalently, formula_3 where formula_4 is the "sample mean"
Proof of the central limit theorem
For a théorem of such fundamental importance to statistics and applied probability, the central limit théorem has a remarkably simple proof using characteristic functions
It is similar to the proof of a (wéak) law of large numbers
For any variabel acak, "Y", with zero méan and unit variance (var("Y") = 1), the characteristic function of "Y" is, by Taylor's theorem, formula_5 Letting "Y""i" be ("X""i" − μ)/σ, the standardised value of "X""i", it is éasy to see that the standardised méan of the observations "X1", "X2", ..., "X"n"" is just formula_6 By simple properties of characteristic functions, the characteristic function of
characteristic functions, the characteristic function of "Z""n" is formula_7 But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit théorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution
Convergence to the limit
If the third central moment E(("X"1 − μ)3) exists and is finite, then the above convergence is uniform and the speed of convergence is at léast on the order of 1/"n"½ (see Berry-Esséen theorem)
Pictures of a distribution being "smoothed out" by summation (showing original distribution and three subseqent convolutions): An equivalent formulation of this limit théorem starts with "A""n" = ("X"1 + ... + "X""n") / "n" which can be interpreted as the méan of a random sample of size "n"
The expected value of "A""n" is μ and the standard deviation is σ / "n"½
If we normalize "A""n" by setting "Z""n" = ("A""n" - μ) / (σ / "n"½), we obtain the same variable "Z""n" as above, and it approaches a standard normal distribution
Note the following apparent "paradox": by adding many independent identically distributed "positive" variables, one gets approximately a normal distribution
But for every normally distributed variable, the probability that it is negative is non-zero
How is it possible to get negative numbers from adding only positives
The réason is simple: the théorem applies to terms centered about the méan
Without that standardization, the distribution would, as intuition suggests, escape away to infinity
Alternative statements of the theorem
The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist)
Thus the central limit théorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions incréases without bound, under the conditions stated above
Since the characteristic function of a convolution is the product of the characteristic functions of the densities involved, the central limit théorem has yet another restatement: the product of the characteristic functions of a number of density functions tends to the characteristic function of the normal density as the number of density functions incréases without bound, under the conditions stated above
An equivalent statement can be made about Fourier transforms, since the characteristic function is essentially a Fourier transform
Let "X""n" be a sequence of independent random variables defined on the same probability space
Assume that "X""n" has finite expected value μ"n" and finite standard deviation σ"n"
We define formula_8 Assume that the third central moments formula_9 are finite for every "n", and that formula_10 (This is the Lyapunov condition)
We again consider the sum "Sn"="X1+...+Xn"
The expected value of "S""n" is "m""n" = ∑"i"=1.."n"μ"i" and its standard deviation is "s""n"
If we normalize "S""n" by setting formula_11 then the distribution of "Z""n" converges towards the standard normal distribution N(0,1) as above
In the same setting and with the same notation as above, we can replace the Lyapunov condition with the following wéaker one: for every ε &gt; 0 formula_12 (where E( "U" : "V" &gt; "c") denotes the conditional expected value: the expected value of "U" given that "V" &gt; "c".)
Then the distribution of the normalized sum "Z""n" converges towards the standard normal distribution N(0,1)
There are some théorems which tréat the case of sums of non-independent variables, for instance the "m-dependent central limit theorem", the "martingale central limit theorem" and the "central limit theorem for mixing processes"
"track these down"
Dina tiori probabiliti, keur nyebutkeun yén dua kajadian independent atawa mandiri dumasar kana pamikiran nu gampang yén pangaweruh kana ayana hiji kajadian lain disababkeun ku ayana pangaruh kamungkinan tina hiji kajadian séjénna
Upamana, keur meunang angka "1" dina sakali ngalungkeun dadu sarta meunang deui angka "1" dina alungan dadu kadua mangrupa conto kajadian mandiri
Hal nu sarupa, waktu urang nyebutkeun dua variabel acak bébas, we intuitively méan that knowing something about the value of one of them does not yield any information about the value of the other
For example, the number appéaring on the upward face of a die the first time it is thrown and that appéaring the second time are independent
If two events "A" and "B" are independent, then the conditional probability of "A" given "B" is the same as the "unconditional" (or "marginal") probability of "A", i.e., formula_1 There are at léast two réasons why this statement is not taken to be the definition of independence: (1) the two events "A" and "B" do not play symmetrical roles in this statement, and (2)
roles in this statement, and (2) problems arise with this statement when events of probability 0 are involved
When one recalls that the conditional probability P("A" | "B") is given by formula_2 one sees that the statement above is equivalent to formula_3 Here "A" ∩ "B" is the intersection of "A" and "B", i.e., it is the event that both events "A" and "B" occur
Thus we could say: Thus the standard definition says: Two events "A" and "B" are independent iff P("A" ∩ "B")=P("A")P("B")
More generally, and collection of events—possibly more than just two of them—are mutually independent precisely if for any finite subset "A"1, ..., "A""n" of the collection we have formula_4 This is called the "multiplication rule" for independent events
If any two of a collection of random variables are independent, they may nonetheless fail to be mutually independent; this is called pairwise independence
Independent random variables
Two random variables "X" and "Y" are independent iff for any numbers "a" and "b" the events ["X" ≤ "a"] and ["Y" ∈ "b"] are independent events as defined above