izzako/sundanese-pixelgpt huggingface dataset:
There are several distinct laws and legal traditions, and éach jurisdiction has its own set of laws and its own legal system
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Rumus pikeun ngitung populasi varian: formula_1 Rumus pikeun ngitung unbiased estimasi populasi varian tina sampel "terhingga" nyaéta: formula_2 Cara ngitung bakal leuwih gampang kaharti dina tabel di handap ieu di mana nilai méan = 8
Catetan: Ngitung varian sacara lengkep: 338 = [52 + 72 + 82 + 102 + 102]&lt;br&gt; 40 = [5 + 7 + 8 + 10 + 10] Algoritma Algoritma sederhana keur ngitung varian saperti di handap ieu: Algoritma Algoritma séjén keur jumlah angka anu loba
"formula") nyaéta jalan ringkes keur méré gambaran informasi (saperti dina matematik atawa rumus kimia) atawa "hubungan umum antar kuantitas"
Salah sahiji nu kawentar nyaéta rumus Albert Einstein E = m × c 2 (tempo Special relativity)
Dina kamungkinan jeung statistik, sebaran log-normal nyaéta probability distribution nu raket hubunganna jeung sebaran normal: lamun "X" mangrupa variabel acak dina sebaran normal, maka exp("X") mibanda sebaran log-normal
Dina basa séjén: variabel natural logarithm sebaran log-normal mibanda sebaran normal
"Log-normal" ogé disebut "log normal" atawa "lognormal"
Variable bisa dimodélkeun salaku log-normal lamun mangrupa product hasil kali tina sababaraha faktor bébas
Conto tipena nyaéta angka ti "return rate" bursa efek dina waktu nu lila: bisa dianggap salalu produk harian "return rate"
Sebaran log-normal mibanda probability density function formula_1 keur "x" &gt; 0, nu mana μ and σ nyaéta mean jeung simpangan baku tina variabel logaritma
Nilai ekspektasi nyaéta formula_2 jeung varian nyaéta formula_3
Hubungan geometrik mean jeung geometrik simpangan baku
Sebaran log-normal, geometric mean, jeung geometri simpangan baku mangrupa hal nu pakait
Dina kasus , géometric méan sarua jeung formula_4 sarta géometric simpangan baku sarua jeung formula_5
Lamun sampel data nu ditangtukeun asalna ti populasi sebaran log-normal, géometric méan jeung géometric simpangan baku bisa dipaké keur nga-estimasi confidence interval ku jalan arithmetic méan jeung simpangan baku nu digunakeun keur nga-estimasi confidence interval dina sebaran normal
nu mana géometric méan formula_6 jeung géometri simpangan baku formula_7
Power uji statistik nyaéta tes probabilitas nu bakal nolak kasalahan null hypothesis, atawa dina basa séjén moal maké Type II error
Beuki luhur power, beuki gedé kamungkinan Type II error nurun tur sabalikna
Probablitas Type II error dumasar kana gedéna negatif kasalahan (β)
Sanajan kitu power sarua jeung 1 − β. Analisa power séjénna bisa saméméh ("priori") atawa sanggeus ("post hoc") data dikumpulkeun
A priori power analysis is conducted prior to the conducting of reséarch and is typically used to determine an appropriate sample size to achieve adequate power
Post-hoc power analysis is conducted after a study has been conducted and uses the obtained sample size and effect size to determine what the power was in the study assuming the effect size in the sample size is equal to the population effect size
Statistical tests attempt to use data from samples to determine if differences or similarities exist in a population
For example, to test the null hypothesis that the mean scores of men and women on a test do not differ, samples of men and women are drawn, the test is administered to them, and the méan score of one group is compared to that of the other group using a statistical test
The power of the test is the probability that the test will find a statistically significant difference between men and women, as a function of the size of the true difference between those two populations
Despite the use of random samples, which will tend to mirror the population due to mathematical properties such as the central limit theorem, there is always a chance that the samples will appéar to support or refute a tested hypothesis when the réality is the opposite
This risk is quantified as the power of the test and as the statistical significance level used for the test
Statistical power depends on: A significance criterion is a statement of how unlikely a result must be, if the null hypothesis is true, to be considered significant
The most commonly used criteria are probabilities of 0.05 (5%, 1 in 20), 0.01 (1%, 1 in 100), and 0.001 (0.1%, 1 in 1000)
If the criterion is 0.05, the probability of the difference must be less than 0.05, and so on
One way to incréase the power of a test is to incréase (that is, wéaken) the significance level
This incréases the chance of obtaining a statistically significant result (rejecting the null hypothesis) when the null hypothesis is false, that is, reduces the risk of a Type II error
But it also incréases the risk of obtaining a statistically significant result when the null hypothesis is in fact true; that is, it incréases the risk of a Type I error
Calculating the power requires first specifying the effect size you want to detect
The gréater the effect size, the gréater the power
Sensitivity can be incréased by using statistical controls, by incréasing the reliability of méasures (as in psychometric reliability), and by incréasing the size of the sample
Incréasing sample size is the most commonly used method for incréasing statistical power
Although there are no formal standards for power, most reséarchers who assess the power of their tests use 0.80 as a standard for adequacy
A common misconception by those new to statistical power is that power is a property of a study or experiment
In réality any statistical result that has a p-value has an associated power
For example, in the context of a single multiple regression, there will be a different level of statistical power associated with the overall r-square and for éach of the regression coefficients
When determining an appropriate sample size for a planned study, it is important to consider that power will vary across the different hypotheses
There are times when the recommendations of power analysis regarding sample size will be inadequate
Power analysis is appropriate when the concern is with the correct acceptance or rejection of a null hypothesis
In many contexts, the issue is less about determining if there is or is not a difference but rather with getting a more refined estimate of the population effect size
For example, if we were expecting a population correlation between intelligence and job performance of around .50, a sample size of 20 will give us approximately 80% power (alpha = .05, two-tail)
However, in doing this study we are probably more interested in knowing whether the correlation is .30 or .60 or .50
In this context we would need a much larger sample size in order to reduce the confidence interval of our estimate to a range that is acceptable for our purposes
These and other considerations often result in the true but somewhat simplistic recommendation that when it comes to sample size, "More is better!"
However, huge sample sizes can léad to statistical tests becoming so powerful that the null hypothesis is always rejected for réal data
This is a problem in studies of differential item functioning
Funding agencies, ethics boards and reséarch review panels frequently request that a reséarcher perform a power analysis
The argument is that if a study is inadequately powered, there is no point in completing the reséarch
Salah sahiji hal nu pakait jeung meupeuskeun masalah nyaéta nyieun kaputusan nu hadé dumasar kana hipotesa nu teu pasti ngaliwatan panalungtikan
Tes hipotesa statistik, atawa leuwih ilahar disingkat, "tes hipotesa", nyaéta algoritma keur nangtukeun alternatip (keur atawa lawan hipotesa) nu ngaminimalkeun resiko kasalahan
Kaca ieu ngajéntrékeun nu remen digunakeun dina tes hipotesa
Tina panempo Bayesian, mangrupa hal nu pas keur maké tes hipotesa dina kasus teori kaputusan normatip (hususna dina masalah pamilihan model) sarta mungkin keur ngumpulkeun bukti pitulung hipotesa saperti maké konsép rasio likelihood nu disebut Bayes factors
Aya sababaraha hal nu perlu disiapkeun saméméh nalungtik data
Sanggeus data aya, uji statistik diitung sarta ditangtukeun dina jero daerah kritis
Lamun uji statistik aya dina jero daerah kritis, maka kasimpulanna nyaéta panalungtik geus milih antara dua logika alternatif ieu
Dina conto bisa disebutkeun: respon observasi kana perlakuan signifikan sacara statistik
Lamun tes statistik di luar daerah kritis, ngan hiji kasimpulan nyaéta Ieu teu sarua minangka kajadian keur hipotesa
Hal ieu teu bisa dipaké minangka alesan, sabab kurangna kajadian tinimbang hipotesa
Dumasar kana hal ieu, kamajuan pananglutikan sacara statistik ku ngurangan kasalahan, lain ku "manggihkeun bebeneran"
falsifiability -- tiori statistik -- statistik terapan -- null hypothesis
Dina tes hipotesa statistik, Type II error miboga gagalna keur nolak null hypothesis nu salah (contona
kasalahan narima hypothesis nu salah)
Simbol keur probabiliti Type II error nyaéta formula_1 (beta)
Power tes statistik dihartikeun salaku formula_2
Tes nu mibanda sensitivity luhur bakal miboga Type II errors saeutik
Tempo ogé Type I error
Dina tes hipotesa statistik, Type I error nyaéta ditolakna null hypothesis lamun bener, dina basa séjén manggihkeun hasil nu miboga statistical significance dina waktu bener-bener kajadian
Hiji tes nu mibanda specificity luhur bakal saeutik ngabogaan "Type I errors"
Simbol keur probabiliti Type I error nyaéta α (alpha) sarta kadang-kadang dipaké salaku "ukuran" tes
Tempo Type II error
Pulitik nyaéta prosés sarta cara kelompok pikeun nyieun kaputusan
Sanajan ilahar dipaké keur pamaréntah, sifat pulitik ogé ditalungtik di pausahaan, akademik, agama sarta institusi séjén
Élmu pulitik nyaéta élmu ngeunaan paripolah politik, uji akurasi sarta pamakéan kakuatan, contona kamampuan keur ngaruntagkeun lawan
Test-t nyaéta unggal tes hipotesa statistik nu mana tes statistik mibanda Sebaran-t student lamun null hypothesis bener
Dina matematika, probability density function dipaké keur ngagambarkeun probability distribution di watesan integrals
Lamun probability distribution mibanda densiti "f"("x"), saterusna interval tak terhingga ["x", "x" + d"x"] mibanda probabiliti "f"("x") d"x"
Probability density function bisa ogé ditempo tina versi "smoothed out" histogram: if one empirically méasures values of a variabel acak repéatedly and produces a histogram depicting relative frequencies of output ranges, then this histogram will resemble the random variable's probability density (assuming that the variable is sampled sufficiently often and the output ranges are sufficiently narrow)
Formally, a probability distribution has density "f"("x") if "f"("x") is a non-negative Lebesgue-integrable function R → R such that the probability of the interval ["a", "b"] is given by formula_1 for any two numbers "a" and "b"
This implies that the total integral of "f" must be 1
Conversely, any non-negative Lebesgue-integrable function with total integral 1 is the probability density of a suitably defined probability distribution
Contona, sebaran seragam dina interval [0,1] mibanda probabiliti densiti "f"("x") = 1 keur 0 ≤ "x" ≤ 1 jeung nol dimamana
Standar sebaran normal mibanda probabiliti densiti formula_2
Lamun variabel acak "X" dibérékeun sarta distribusina kaasup kana fungsi probabiliti densiti "f"("x"), mangka nilai ekspektasi "X" (lamun éta aya) bisa diitung ku formula_3 Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution, even though it has no discrete component, i.e., does not assign positive probability to any individual point
A distribution has a density function if and only if its cumulative distribution function "F"("x") is absolutely continuous
In this case, "F" is almost everywhere differentiable, and its derivative can be used as probability density
If a probability distribution admits a density, then the probability of every one-point set {"a"} is zero
It is a common mistake to think of "f"("a") as the probability of {"a"}, but this is incorrect; in fact, "f"("a") will often be bigger than 1 - consider a random variable with a uniform distribution between 0 and 1/2
Dua densiti "f" jeung "g" for the same distribution can only differ on a set of Lebesgue measure zero
Dina statistik, prediksi interval ngahasilkeun hubungan nu sarua dina observasi nu bakal datang yén interval kapercayaan ngalahirkeun paraméter populasi nu teu ka-observasi
Anggap sampel dicokot tina populasi sebaran normal
Populasi mean jeung simpangan baku teu dipikanyaho sajaba duanana bisa di-estimasi dumasar kana sampel
Hal ieu diperlukeun keur prediksi observasi saterusna
Anggap "n" jadi ukuran sampel; anggap μ jeung σ populasi méan jeung simpangan baku nu teu katalungtik
Anggap "X"1, ..., "X""n", jadi sampel; anggap "X""n"+1 observasi saterusna nu bakal diprediksi
Maka formula_1 jeung formula_2 Saterusna geus ilahar ditunjukkeun ku formula_3 mibanda sebaran-t student nu mibanda "n" − 1 tingkat kabébasan
Akibatna formula_4 nu mana "A" nyaéta percentil 100(1 − (p/2))th ti sebaran-t student nu mibanda "n" − 1 tingkat kabébasan
nu mana angka formula_5 nyaéta the titiktungtung tina 100p% prediksi interval keur "X""n"+1
Keur satiap positip integer formula_1, sebaran chi-kuadrat nu mibanda "k" tingkat kabebasan nyaéta probability distribution variabel acak formula_2 nu mana "Z"1, ..., "Z""k" mangrupa variabel normal bebas, masing-masing nilai ekspektasi 0 jeung varian 1
Sebaran ieu biasa ditulis formula_3 Lamun formula_4 watesan linier homogen bébas ditumpukeun dina ieu variabel, kayaan sebaran formula_5 dina watesan ieu nyaéta formula_6, dipastikeun salaku watesan "tingkat kabebasan"
Characteristic function sebaran Chi-kuadrat nyaéta formula_7 Sebaran chi-kuadrat mibanda aplikasi numeris dina kaputusan statistik, contona dina tes chi-kuadrat jeung estimasi varian
Ieu bisa diasupkeun kana masalah estimasi méan dina populasi sebaran normal jeung masalah estimasi slope dina garis regression ku aturan dina sebaran-t student
Ieu diasupkeun kana sakabéh masalah analisa varian ku aturan dina sebaran-F, nu mangrupa sebaran perbandingan dua chi-kuadrat variabel acak
Rumus probability density function nyaéta formula_8 jeung "p""k"("x") = 0 keur "x"≤0
Di dieu Γ ngalambangkeun fungsi gamma
Lamun formula_9, saterusna formula_1 nuju ka takterhingga, sebaran formula_5 nuju ka normal
Sanajan kitu, kacenderunganna lalaunan (skewness nyaéta formula_12 jeung kurtosis nyaéta formula_13) sarta dua transpormasi umumna diperhatoskeun, unggal pamarekannormal leuwih gancang tinimbang formula_5 sorangan: Fisher nembongkeun yén formula_15 ngadeukeutan sebaran normal nu mibanda méan formula_16 jeung unit varian
Wilson and Hilferty dina taun 1931 nembongkeun yén formula_17 nyaéta pamarekansebaran normal nu mibanda méan formula_18 jeung varian formula_19
Nilai ekspektasi tina variabel random mibanda sebaran chi-kuadrat nu mibanda "k" tingkat kabébasan "k" jeung varian nyaéta 2"k"
Median dina ieu kaayaan dideukeutan ku formula_20 Catetan yén 2 tingkat kabébasan nuju kana sebaran eksponensial
Sebaran chi-kuadrat dina kasus husus nyaéta sebaran gamma
Tempo Teorema Cochran
Dina statistika, téoréma Cochran digunakeun dina analisis varian
Anggap "U"1, ..., "U""n" mangrupa standar variabel random bebas nu kasebar normal, sarta dina bentuk identitas formula_1 bisa dituliskeun yén unggal "Q""i" nyaéta jumlah kuadrat kombinasi liniér tina "U"
Mangka lamun formula_2 nu mana "r""i" mangrupa rangking tina "Q""i", téorema Cochran nangtukeun yén "Q""i" bébas sarta "Q""i" mibanda sebaran chi-kuadrat nu mibanda tingkat kabebasan "r""i"
Téoréma Cochran mangrupa konversi téoréma Fisher
Lamun "X"1, ..., "X""n" mangrupa variabel random bébas nu kasebar normal mibanda méan μ sarta simpangan baku σ mangka formula_3 mangrupa standar normal keur unggal "i"
Ieu mungkin keur nulis formula_4 (di dieu, jumlahna ti 1 nepi ka "n", dumasar kana observasi)
Keur nempo ieu identitas, kalikeun ku formula_5 sarta catet yen formula_6 sarta legaan keur manggihkeun formula_7 Watesan katilu sarua jeung nol sabab ieu angger kana waktu formula_8 sarta watesan kadua ngan watesan "n" identik nu ditambahkeun babarengan
Kombinasi di luhur ngahasilkeun (sarta dibagi ku σ2), urang mibanda: formula_9 Ayeuna rengking "Q"2 ngan 1 (ieu mangrupa kuadrat tina hiji kombinasi linier variabel normal standar)
Rengking "Q"1 bisa ditembongkuen jadi "n" − 1, sarta kondisi téorema Cochran kapanggih
Téorema Cochran netepkeun yén "Q"1 and "Q"2 mangrupa bébas, mibanda sebaran chi-kuadrat "n" − 1 sarta 1 tingkat kabébasan
Ieu nembongkeun yén sampel méan sarta sampel varian bébas; sarta formula_10 Keur "estimasi" varian &amp;sigma2, hiji éstimator nu biasa digunakeun nyaéta formula_11
Téorema Cochran nembongkeun yen formula_12 nu nembongkeun yén nilai ekspektasi formula_13 nyaéta σ2"n"/("n" − 1)
Dua sebaran ieu mangrupa "proporsi" kana varian sabenerne tapi teu dipikanyaho σ2; mangka ieu rasio mangrupa σ2 bébas sabab duana bébas, mangka urang miboga formula_14 nu mana "F"1,"n" mangrupa sebaran-F nu mibanda 1 sarta "n" tingkat kabébasan (tempo ogé sebaran-t student)
Dina téori probabilitas jeung statistika, skewness nyaéta ukuran kateu-simetrian sebaran probabilitas tina nilai-real variabel acak
Sacara kasar bisa disebutkeun, sebaran mibanda "skew" positip lamun nilai panjang "tail" positip sarta skew negatip lamun nilai panjang "tail" negatip
Skewness, standardized moment nu katilu, dihartikeun ku μ3 / σ3, nu mana μ3 mangrupa moment about the mean nu katilu sarta σ mangrupa simpangan baku
"Skewness" tina random variable "X" kadangkala ngalambangkeun "Skew"["X"]
Keur nilai sampel "N" sampel "skewness" nyaéta Σ"i"("x""i" − μ)3 / "N"σ3, where "x""i" mangrupa nilai "i"th jeung μ mangrupa mean
Lamun "Y" nyaéta jumlah "n" independent variabel random, dina distribusi nu sarua salaku "X", saterusna ditempokeun yén "Skew" ["Y"] = Skew["X"] / √"n"
Sampel nu asalna tina populasi, persamaan keur populasi "skewness" nyaéta biased estimator tina populasi "skewness"
Unbiased estimator "skewness" nyaéta formula_1 nu mana σ mangrupa sample simpangan baku sarta μ mangrupa sampel méan
Tempo oge: mean, varian, kurtosis, cumulant
Dina tiori probabilitas jeung statistika, kurtosis hartina ukuran "posisi puncak" tina sebaran probabilitas niléy-nyata/ril variabel acak
Standardized moment kaopat dihartikeun minangka μ4 / σ4, nu mana μ4 mangrupa momen mean kaopat jeung σ nyaéta simpangan baku
Hal ieu kadangkala dipaké keur ngahartikeun kurtosis dina pagawéan saméméhna, tapi teu dipaké dina definisi di dieu
Kurtosis leuwih ilahar dihartikeun ku μ4 / σ4 − 3
Minus 3 di tungtung persamaan éta nerangkeun yén koreksi dijieun keur ngajadikeun kurtosis sebaran normal sarua jeung nol
Alesan séjénna némbongkeun yén kurtosis mangrupa jumlah tina variabel random
Lamun "Y" jumlah tina "n" independent variabel random, sakabéh sebaranna sarua nya éta "X", saterusna Kurt["Y"] = Kurt["X"] / "n", sabalikna ieu rumus bakal leuwih pajuriwet lamun kurtosis dihartikeun ku μ4 / σ4
Sebaran normal mibanda kurtosis sarua jeung nol (sebaran nu mibanda niléy kurtosis sarua jeung nol disebut "mesokurtic")
Sebaran nu mibanda kurtosis positip disebut "leptokurtic", sarta lamun negatif disebut "platykurtic"
Keur niléy sampel "N", sampel kurtosis nyaéta Σ"i"("x""i" − μ)4 / "N"σ4 − 3, nu mana "x""i" nyaéta nilai "i"th jeung μ nyaéta mean
Dina kaayaan bagéan-susunan sampel tina populasi, sampel kurtosis di luhur mangrupa biased estimator ti populasi kurtosis
Unbiased estimator tina populasi kurtosis nyaéta formula_1 nu mana σ nyaéta simpangan baku sampel jeung μ nya éta sampel mean
Tempo oge: mean, varian, skewness
Dina matematik, fungsi distribusi kumulatip (disingkat cdf) ngajelaskeun probability distribution ti sakabéh nilai-real variabel acak, "X"
Keur satiap réal number "x", cdf dirumuskeun ku formula_1 nu mana sisi beulah katuhu ngagambarkeun probabilitas nu mana variabel acak "X" dicokot tina nilai nu kurang tina atawa sarua jeung "x"
Probabilitas "X" aya dina interval ("a", "b"] nyaéta "F"("b") − "F"("a") lamun "a" &lt; "b"
Geus ilahar ngagunakeun huruf "F" gedé keur fungsi sebaran kumulatif, nu jelas béda jeung huruf "f" leutik nu dipaké keur fungsi dénsitas probabilitas jeung probability mass function
Fungsi sebaran kumulatif X bisa dihartikeun dina watesan fungsi dénsitas probabilitas "f" saperti kieu: formula_2 Note that in the definition above, the "less or equal" sign, '≤' is a convention, but it is an important and universally used one
The proper use of tables of the Binomial and Poisson distributions depend upon this convention
Moréover, important formulas like Levy's inversion formula for the characteristic function also rely on the "less or equal" formulation
Every cumulative distribution function "F" is (not necessarily strictly) monotone increasing and right-continuous
Furthermore, we have formula_3 Every function with these four properties is a cdf
The properties imply that all CDFs are càdlàg functions
If "X" is a discrete random variable, then it attains values "x"1, "x"2, ... with probability "p"i = P("x"i), and the cdf of "X" will be discontinuous at the points "x""i" and constant in between: formula_4 If the CDF "F" of "X" is continuous, then "X" is a continuous random variable; if furthermore "F" is absolutely continuous, then there exists a Lebesgue-integrable function "f"("x")
there exists a Lebesgue-integrable function "f"("x") such that formula_5 for all réal numbers "a" and "b"
(The first of the two equalities displayed above would not be correct in general if we had not said that the distribution is continuous
Continuity of the distribution implies that P("X" = "a") = P("X" = "b") = 0, so the difference between "&lt;" and "≤" céases to be important in this context.)
The function "f" is equal to the derivative of "F" almost everywhere, and it is called the probability density function of the distribution of "X"
The "point probability" that "X" is exactly "b" can be found as formula_6 Kolmogorov-Smirnov and Kuiper's tests
The Kolmogorov-Smirnov test is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an idéal distribution
The closely related Kuiper's test (pronounced ) is useful if the domain of the distribution is cyclic as in day of the week
For instance we might use Kuiper's test to see if the number of tornadoes varies during the yéar or if sales of a product vary by day of the week or day of the month
Complementary cumulative distribution function
Sometimes, it is useful to study the opposite question and ask how often the random variable is "above" a particular level
This is called the complementary cumulative distribution function (ccdf), defined as formula_7
In survival analysis, formula_8 is called the survival function and denoted formula_9
As an example, suppose "X" is uniformly distributed on the unit interval [0, 1]
Then the CDF of X is given by formula_10 Take another example, suppose "X" takes only the discrete values 0 and 1, with equal probability
Then the CDF of X is given by formula_11 Inverse
If the cdf "F" is strictly incréasing and continuous then formula_12 is the unique réal number formula_13 such that formula_14
Unfortunately, the distribution does not, in general, have an inverse
One may define, for formula_15, formula_16
Example 1: The median is formula_17
Example 2: Put formula_18
Then we call formula_19 the 95th percentile
The inverse of the cdf is called the quantile function
Dina matématik, fungsi béta teu lengkep dihartikeun dina bentuk integral, formula_1 mibanda "a" &gt; 0 jeung "b" &gt; 0. formula_2 formula_3 Sifat
"(Loba sifat séjénna nu bisa dituliskeun di dieu)."
Dina statistika, sebaran-F hartina sebaran probabilitas tina variabel acak dina bentuk formula_1 nu mana Sebaran-"F" sering dipaké salaku sebaran null dina tes statistik, husuna dina uji rasio likelihood, diharapkeun leuwih hadé dina analisis varian; tempo uji-F. Lamun "d""2" &gt; 2, saterusna méan "d"2/("d"2 − 2) jeung lamun "d""2" &gt; 4 maka varian dirumuskeun ku formula_2 Sebaran-F salawasna aya dina rentang nol nepi ka "takterhingga";
dina rentang nol nepi ka "takterhingga"; lamun "d""1" &gt; 2, mangrupa unimodal mibanda mode formula_3
Dina probability theory jeung statistik, "k"th standardized moment tina probability distribution nyaéta μk/σk, nu mana μk mangrupa "k"th momen mean jeung σ mangrupa simpangan baku
Standardized moments katilu jeung kaopat digunakeun keur ngartikeun skewness jeung kurtosis
Informasi saterusna ngeunaan "momen", bisa konsultasi ka Wolfram Reséarch's mathworld website
Momen méan "k"th (atawa "k"th central moment) nilai-réal variabel acak "X" nyaéta kuantitas E[("X" − E["X"])"k"], nu mana E mangrupa operator ekspektasi
Sababaraha variabel random teu mibanda mean, dina kasus momen méan teu bisa dihartikeun
Momen méan "k"th salawasna dilambangkeun ku μ"k"
Keur continuous univariate probability distribution nu mibanda probability density function "f"("x") momen méan μ nyaéta formula_1 Kadangkala kacida gampangna keur koversi momen asli ka momen méan
Persamaan umum keur konversi momen asli orde-nth ka momen méan nyaéta formula_2 nu mana "m" mangrupa méan sebaran, sarta momen asli dirumuskeun ku formula_3 Momen méan kahiji nyaéta nol
Momen méan kadua disebut varian, biasa dilambangkeun ku σ2, nu mana σ ngawakilan simpangan baku
Momen méan katilu jeung kaopat ngartikeun standardized moment sacara "berurutan" dipaké keur ngartikeun skewness jeung kurtosis
moment (matematik), cumulant
Cumulants sebaran probabiliti
Dina probability theory jeung statistik, cumulants κ"n" tiprobability distribution dirumuskeun ku formula_1 nu mana "X" nyaéta unggal variabel random sebaran probabiliti is the one whose cumulants are taken
In other words, κ"n"/"n"
is the "n"th coefficient in the power series representation of the logarithm of the moment-generating function
The logarithm of the moment-generating function is therefore called the cumulant-generating function
The "problem of cumulants" attempts to recover a probability distribution from its sequence of cumulants
In some cases no solution exists; in some cases a unique solution exists; in some cases more than one solution exists
Some properties of cumulants
Invariance and equivariance
The first cumulant is shift-equivariant; all of the others are shift-invariant
To state this less tersely, denote by κ"n"("X") the "n"th cumulant of the probability distribution of the random variable "X"
The statement is that if "c" is constant then κ1("X" + "c") = κ1("X") + "c" and κ"n"("X" + "c") = κ"n"("X") for "n"≥ 2, i.e., "c" is added to the first cumulant, but all higher cumulants are unchanged
The "n"th cumulant is homogenéous of degree "n", i.e
if "c" is any constant, then formula_2 Additivity
If "X" and "Y" are independent random variables then κ"n"("X" + "Y") = κ"n"("X") + κ"n"("Y")
Cumulants and moments
The cumulants are related to the moments by the following recursion formula: formula_3 The "n"th moment μ′"n" is an "n"th-degree polynomial in the first "n" cumulants, thus: formula_4 formula_5 formula_6 formula_7 formula_8 formula_9 The "prime" distinguishes the moments μ′"n" from the central moments μ"n"
To express the "central" moments as functions of the cumulants, just drop from these polynomials all terms in which κ1 appéars as a factor
The coefficients are precisely those that occur in Faà di Bruno's formula
Cumulants and set-partitions
These polynomials have a remarkable combinatorial interpretation: the coefficients count certain partitions of sets
A general form of these polynomials is formula_10 where Thus éach monomial is a constant times a product of cumulants in which the sum of the indices is "n" (e.g., in the term κ3 κ22 κ1, the sum of the indices is 3 + 2 + 2 + 1 = 8; this appéars in the polynomial that expresses the 8th moment as a function
the 8th moment as a function of the first eight cumulants)
A partition of the integer "n" corresponds to éach term
The "coefficient" in éach term is the number of partitions of a set of "n" members that collapse to that partition of the integer "n" when the members of the set become indistinguishable
Cumulants of particular probability distributions
Cumulant sebaran normal mibanda nilai ekspektasi μ sarta varian σ2 mangrupa κ1 = μ, κ2 = σ2, sarta κ"n" = 0 keur "n" &gt; 2
Sakabéh cumulant sebaran Poisson sarua jeung nilai ekspektasi
A distribution with arbitrary given cumulants κ"n" can be approximated through the Gram-Charlier or Edgeworth series
The joint cumulant of several random variables "X"1, ..., "X""n" is formula_11 where π runs through the list of all partitions of { 1, ..., "n" }, and "B" runs through the list of all block of the partition π
For example, formula_12 The joint cumulant of just one random variable is its expected value, and that of two random variables is their kovarian
If some of the random variables are idependent of all of the others, then the joint cumulant is zero
If all "n" random variables are the same, then the joint cumulant is the "n"th ordinary cumulant
The combinatorial méaning of the expression of moments in terms of cumulants is éasier to understand than that of cumulants in terms of moments: formula_13 where κ("X""B") is the joint cumulant of those among the random variables "X"1, ..., "X""n" whose indices are included in the block "B"
For example: formula_14 Conditional cumulants
The law of total expectation and the law of total variance generalize naturally to conditional cumulants
The case "n" = 3, expressed in the language of (central) moments rather than that of cumulants, says formula_15 The general result stated below first appéared in 1969 in "The Calculation of Cumulants via Conditioning" by David R. Brillinger in volume 21 of "Annals of the Institute of Statistical Mathematics", pages 215-218
In general, we have formula_16 where History
Cumulants were first introduced by the Danish astronomer, actuary, mathematician, and statistician Thorvald N. Thiele (1838 - 1910) in 1889
Thiele called them "half-invariants"
They were first called "cumulants" in a 1931 paper, "The derivation of the pattern formulae of two-way partitions from those of simpler patterns", Proceedings of the London Mathematical Society, Series 2, v. 33, pp
195–208, by the gréat statistical geneticist Sir Ronald Fisher and the statistician John Wishart, eponym of the Sebaran Wishart
The historian Stephen Stigler has said that the name "cumulant" was suggested to Fisher in a letter from Harold Hotelling
In another paper published in 1929, Fisher had called them "cumulative moment functions"
More generally, the cumulants of a sequence { "m""n" : "n" = 1, 2, 3, ... }, not necessarily the moments of any probability distribution, are given by formula_17 where the values of κ"n" for "n" = 1, 2, 3, ... are found "formally", i.e., by algebra alone, in disregard of questions of whether any series converges
All of the difficulties of the "problem of cumulants" are absent when one works "formally"
The simplest example is that the second cumulant of a probability distribution must always be nonnegative, and is zero only if all of the higher cumulants are zero
"Formal" cumulants are subject to no such constraints
One well-known example
In combinatorics, the "n"th Bell number is the number of partitions of a set of size "n"
All of the cumulants of the sequence of Bell numbers are equal to 1
The Bell numbers are the moments of the Poisson distribution with nilai ekspektasi 1
Cumulants of a polynomial sequence of binomial type
For any sequence { κ"n" : "n" = 1, 2, 3, ... } of scalars in a field of characteristic zero, being considered formal cumulants, there is a corresponding sequence { μ ′ : "n" = 1, 2, 3, ... } of formal moments, given by the polynomials above
For those polynomials, construct a polynomial sequence in the following way
Out the polynomial formula_18 make a new polynomial in these plus one additional variable "x": formula_19 ... and generalize the pattern
The pattern is that the numbers of blocks in the aforementioned partitions are the exponents on "x"
éach coefficient is a polynomial in the cumulants; these are the Bell polynomials, named after Eric Temple Bell
This sequence of polynomials is of binomial type
In fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of formal cumulants
Moment about the mean
Frase "tingkat kebebasan" digunakeun dina 3 cabang elmu anu béda nyaéta: dina fisika jeung physical chemistry, dina mechanical jeung aeronautical engineering, sarta dina statistik
Tiluannana dihubungkeun sacara sajarah jeung ngaliwatan matematik nyaéta konsép dimensionality, sanajan tiluanana teu identik
Fisika jeung kimia
Dina fisika jeung kimia, unggal mode mandiri tempat hiji partikel atawa sistim bisa gerak atawa ngarah disebut hiji darajat kabébasan
For a roughly dumbell-shaped hydrogen molecule, three such modes would be rotation (twirling), translation (hurtling through space) and vibration (the two dumbbell "balls" bouncing together and apart)
According to the Equipartition Theorem of thermodynamics, in case of thermal equilibrium éach degree of freedom in every particle of a system will contain the same energy on average (equal to kT, the temperature of the system multiplied by the fundamental Boltzmann constant)
However, thermal equilibrium can only be réached among interacting particles, a process called thermalization
According to quantum mechanics and more specifically Heisenberg's uncertainty principle, the amount of energy within any degree of freedom is never zero, but is always at léast equal to the zero-point energy for that mode
In mechanical and aeronautical engineering, degrees of freedom (DOF) describes flexibility of motion
A mechanism that has complete freedom of motion (even if only in a limited aréa, or envelope) has six degrees of freedom
Three modes are translation - the ability to move in éach of three dimensions
Three are rotation, or the ability to change angle around three perpendicular axes
To put it in simpler terms, éach of the following is one degree of freedom: A mechanism that can (for instance) be raised and lowered, which has a pivoting héad that can tilt forward or back, left or right, can be described as having 3 degrees of freedom (colloquially, 3DOF)
Dina statistik, tingkat kabébasan nyaéta statistical parameter anu penting dina probability distribution
Contona dipaké dina sebaran chi-kuadrat, sebaran-F, sebaran-t student, jeung sebaran beta
Tempo Uji kuadrat-chi Pearson jeung analisa varian keur informasi saterusna
Dina sebaran anu geus ilahar dipaké, tingkat kabébasan ngan nyokot integer (umumna kurang ti hiji)
Sacara matematik diidinan keur ngabagi-bagi tingkat kabébasan, antukna meunang hasil itungan nu nyugemakeun
Algorit Another way to classify algorithms is by implementation
A recursive algorithm is one that invokes (makes reference to) itself repéatedly until a certain condition matches, which is a method common to functional programming
Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time
Those computers are sometimes called serial computers
An algorithm designed for such an environment is called a serial algorithm, as opposed to parallel algorithms, which take advantage of computer architectures where several processors can work on a problem at the same time
The various heuristic algorithm would probably also fall into this category, as their name (e.g
a genetic algorithm) describes its implementation
A list of algorithms discussed in Wikipedia is available
Komputasiat, 2007 September 07 Penyéarah Tiga Fasa Terkendali Berdasarkan semikonduktor yang digunakan dan variasi tegangan keluarannya, penyéarah tiga-fasa dapat diklasifikasikan menjadi : • Penyerah tak terkendali
• Penyéarah terkendali
Umumnya senikonduktor penyéarah terkendali menggunakan bahan semikonduktor berupa thyristor, atau menggunakan thyristor dan dioda secara bersamaan
Berdasarkan bahan semikonduktar yang digunakan dan sistem kendalinnya penyéarah tiga-fasa terkendali umumnya dapat dibédakan menjadi : • Half wave Rectifiers • Full wave Rectifiers-Full Controller • Full wave Rectifiers-Semi Controller Hal-hal yang menjadi masalah dalam téhnik penyerahan antara lain adalah trafo penyéarahan, gangguan-gangguan tegangan lebih atau arus lebih yang membahayakan dioda / thyristor, keperluan daya buta untuk beban penyéarahan, harmonisa yang timbul akibat gelombang
penyéarahan, harmonisa yang timbul akibat gelombang non sinus sarta sirkit elektronik pengatur penyalaan
Skema penyéarah terkendali tiga-fasa, masing-masing ditunjukkan pada gambar dibawah ini
1 Penyéarah terkendali
; (a) Half wave rectifiers, (b) Full wave rectifiers-semi controller, (c) Full wave rectifiers-Full controller Gb
2 Gelombang tegangan dan arus penyéarah tiga-fasa jenis Full wave Rectifier-semi controller Dalam aplikasinya, sirkit-sirkit penyéarahan biasanya dilengkapi dengan sirkit
pengatur tambahan seperti pengatur tegangan pembatas arus dan-lain-lain sesuai dengan jenis pemakaiannya
Bidang gerak téhnik penyéarahan meliputi sistem¬-sistem pengatur putaran mesin DC pada mesin cetak kertas, tekstil, mesin las DC, pengisi baterai,sampai pada pengatur tegangan konstan generator sinkron (AVR)
Pada praktikum dipelajari lebih lanjut mengenai karakteristik system penyéarahan khususnya penyéarah tiga fasa gelombang penuh setengah terkendali
Tegangan keluaran penyéarah 1
(b) ditunjukkan pada Gb:2 dan secara matematik dapat dijelaskan sebagai berikut : Pada grup komutasi T1-2-3, karena yang digunakan thyristor, keluarannya tergantung pada sudut penyalaan , sehingga keluarannya : ... (1) Sedangkan pada grup kornutasi D1-2-3
, karena menggunakan dioda, maka keluarannya menjadi : ... (2) Tegangan keluaran total pada beban adalah dengan mensubstitusikan (1) dan (2) diperoleh : "The theory of computation" began éarly in the twentieth century, before modérn electronic computers had been invented
At that time, mathematicians were trying to find which math problems can be solved by simple methods and which cannot
The first step was to define what they méant by a "simple method" for solving a problem
In other words, they needed a formal modél of computation
Several different computational modéls were devised by these éarly reséarchers
One modél, the Turing machine, stores characters on an infinitely long tape, with one square at any given time being scanned by a réad/write héad
Another modél, recursive functions, uses functions and function composition to operate on numbers
The lambda calculus uses a similar approach
Still others, including Markov algorithms and Post systems, use grammar-like rules to operate on strings
All of these formalisms were shown to be equivalent in computational power—that is, any computation that can be performed with one can be performed with any of the others
They are also equivalent in power to the familiar electronic computer, if one pretends that electronic computers have infinite memory
Indeed, it is widely believed that all "proper" formalizations of the concept of algorithm will be equivalent in power to Turing machines; this is known as the Church-Turing thesis
In general, questions of what can be computed by various machines are investigated in computability theory
The théory of computation studies these modéls of general computation, along with the limits of computing: Which problems are (provably) unsolvable by a computer
(See the halting problem and the Post correspondence problem.)
Which problems are solvable by a computer, but require such an enormously long time to compute that the solution is impractical
(See Presburger arithmetic.)
Can it be harder to solve a problem than to check a given solution
(See complexity classes P and NP)
In general, questions concerning the time or space requirements of given problems are investigated in complexity theory
In addition to the general computational modéls, some simpler computational modéls are useful for special, restricted applications
Regular expressions, for example, are used to specify string patterns in UNIX and in some programming languages such as Perl
Another formalism mathematically equivalent to regular expressions, Finite automata are used in circuit design and in some kinds of problem-solving
Context-free grammars are used to specify programming language syntax
Non-deterministic pushdown automata are another formalism equivalent to context-free grammars
Primitive recursive functions are a defined subclass of the recursive functions
Different modéls of computation have the ability to do different tasks
One way to méasure the power of a computational modél is to study the class of formal languages that the modél can generate; this léads to the Chomsky hierarchy of languages
The following table shows some of the classes of problems (or languages, or grammars) that are considered in computability théory (blue) and complexity théory (green)
If class X is a strict subset of Y, then X is shown below Y, with a dark line connecting them
If X is a subset, but it is unknown whether they are equal sets, then the line is lighter and is dotted
This article contains some content from an article by Nancy Tinkham , originally posted on Nupedia
This article is open content
Hidrogéologi ("hidro-" hartina cai, jeung "-géologi" ngartina élmu ngeunaan bumi) nyaéta widang géologi nu nalungtik ngeunaan sebaran jeung gerak cai taneuh dina taneuh jeung batu di kerak Bumi, (ilaharna dina akuifer)
Watesan géohidrologi ogé ilahar dipaké keur ngagantina
Sababaraha bédana antara hidrogéologi jeung géohidrologi nyaéta ahli hidrologi atawa ahli rekayasa ngagunakeun élmu hidrologi kana géologi (géohidrologi), jeung ahli géologi maké élmu géologi kana hidrologi (hidrogéologi)
Hidrogéologi (saperti élmu bumi séjén) mangrupa pangawéruh antar disiplin; hal nu hésé keur mastikeun pakaitna kimia, fisika, biologi kaasup ogé hukum kana taneuh, cai, biosfir jeung kahirupan sosial
Pangaweuh ngeunaan pakaitna gerak cai taneuh jeung géologi kacida ruwetna
Cai taneuh teu salawasna ngalir di handapeun taneuh dumasar kana "topografi permukaan tanah"; cai taneuh ngalir dumasar kana "gradien tekanan" (ngalir ti gradien tekanan tinggi ke rendah) salawasna nuturkeun celah jeung rongga dina pola aliranna
Keur nyaho sacara jéntré hal nu mangaruhan kana sistim multi-komponen nu béda katempona, salawasna merlukeun pangaweruh dina sababaraha widang elmu boh tingkat percobaan atawa teori
Bisa disebutkeun, saterusna leuwih mangrupa bubuka tradisional (panempo nu kurang) kana métodeu jeung catetan ngeunaan hidrologi handapeun taneuh atawa hidrogéologi sederhana
Hidrologi (tina basa Yunani: Yδωρ, hudōr, "cai"; jeung λόγος, logos, "élmu") nyaéta élmu ngeunaan kajadian, distribusi, sarta pindahna cai dina, di jero jeung di luhur Marcapada
Siklus pindahna cai antara térasfir (darat), oséanosfir (laut), jeung atmosfir (udara) disebut daur hidrologi
Aya dua widang utama dina hidrologi
Kahiji hidrologi cai permukaan ("surface-water hydrology") nu museurkeun kana cai dina jeung luhureun taneuh
Conto tina "hidrologi permukaan" nyaéta banjir jeung kasaatan
Nu kadua nyaéta hidrologi cai-taneuh atawa géohidrologi, nu museurkeun kana distribusi jeung pindahna cai di handapeun taneuh (contona, cai taneuh, "groundwater")
Hidrologi cai taneuh penting keur dipaké dina pasokan cai, irigasi jeung rékayasa lingkungan
Catetan yén cai di laut dipisahkeun tina hidrologi sarta dipikawanoh dina istilah oséanografi, sedengkeun cai di atmosfir leuwih diajarkeun dina météorologi
Kaasup dina hidrologi ogé élmu gerak cai sarta "water-borne constituents" — bahan-bahan nu dibawa boh salaku nu leyur atawa dina fase nu misah
A related facet of hydrology is the determination of statistical flow prediction in rivers and stréams
This information is essential to design and evaluation of natural and man-made channels, bridge openings and dams
Stream gage (U.S. Géological Survey terminology) data have been collected and tabulated by the United States Géological Survey for many yéars and much of it is available online for analysis
Hidrologi jadi salah sahiji subyek panalungtingkan jeung rekayasa keur sababaraha milenia
Contona, dina kira-kira 4000 S.M
walungan Nil dibendung keur naekkeun hasil tatanen di wewengkon nu tadina tandus
Kota Mesopotamia ditangtayungan tina banjir ku tembok taneuh di jangkung
Saluran cai diwangun ku Yunani Kuno jeung Romawi Kuno, sedengkeun Sajarah China ngawangun irigasi jeung wangunan keur ngontrol banjir
Sinhale Kuno ngagunakeun hidrologi keur ngawangun Kompleks Irigasi di Sri Lanka Kuno, dikenal salaku nu manggihkeun Lobang Klep nu bisa dipaké keur nyieun tempat panampungan cai nu gedé, saluran cai nu nepi ka kiwari masih kénéh dipake
Marcus Vitruvius, di mimiti abad S.M., ngagambarkeun téori filosofis siklus hidrologi, hujan anu murag di gunung nyerep ka jero taneuh sarta nuju ka walungan sarta usum semi di wewengkon lengkob
Kalawan nyoko pendekatan anu leuwih ilmiah, Léonardo da Vinci sarta Bernard Palissy sacara teuneung mere gambaran nu pas siklus hidrologi
Henteu nepi ka abad ka-17, yén variabel hidrologi mimiti diukur
Pelopor élmu pangaweruh hidrologi modérn nyaéta Pierre Perrault, Edme Mariotte jeung Edmund Halley
Kalawan ngukur curah hujan, "runoff", sarta lega drainase, Perrault méré conto yén curah hujan éta téh mahi keur ngajelaskeun aliran ka walungan Seine
Marriotte ngahijikeun kecepatan sarta "penampang melintang sungai" keur ngitung "debit" walungan Seine
Halley némbongkeun yén penguapan ti Laut Mediterania nyaéta mahi keur ngajelaskeun ngocorna cai walungan ka laut
Advances in the 18th century included the Bernoulli piezometer and Bernoulli's equation, by Daniel Bernoulli, the Pitot tube
The 19th century saw development in groundwater hydrology, including Darcy's law, the Dupuit-Thiem well formula, and Hagen-Poiseuille's capillary flow equation
Rational analyses began to replace empiricism in the 20th century, while governmental agencies began their own hydrological reséarch programs
Of particular importance were Leroy Sherman's unit hydrograph, the infiltration théory of Robert E. Horton, and C.V. Theis's Aquifer test/equation describing well hydraulics
Since the 1950's, hydrology has been approached with a more théoretical basis than in the past, facilitated by advances in the physical understanding of hydrological processes and by the advent of computers and especially Geographic Information Systems (GIS)
The central theme of hydrology is that water moves throughout the éarth through different pathways and at different rates
The most vivid image of this is in the evaporation of water from the océan, which forms clouds
These clouds drift over the land and produce rain
The rainwater flows into lakes, rivers, or aquifers
The water in lakes, rivers, and aquifers then either evaporates back to the atmosphere or eventually flows back to the océan, completing a cycle
Chemical hydrology is the study of the chemical characteristics of water
Ecohydrology is the study of interactions between organisms and the hydrologic cycle
Hydrogeology is the study of the presence and movement of water in aquifers