Meta tags:
Headings (most frequently used words):
random, example, variable, of, variables, equality, definition, distribution, functions, equivalence, contents, examples, measure, theoretic, moments, some, properties, convergence, see, also, references, external, links, standard, case, extensions, discrete, continuous, mixed, type, real, valued, in, almost, sure, practical, difference, between, notions, inline, citations, literature, coin, toss, dice, roll,
Text of the page (most frequently used words):
the (313), displaystyle (220), random (190), #variable (99), probability (94), and (72), distribution (60), variables (60), for (54), function (52), that (46), space (44), can (44), are (41), omega (41), real (38), this (37), edit (34), which (31), measure (30), operatorname (30), continuous (27), then (26), values (26), such (26), set (25), valued (25), numbers (24), example (24), mathbb (24), functions (24), measurable (23), frac (23), leq (21), value (21), text (20), with (20), statistics (19), from (19), equal (18), sqrt (18), all (17), not (17), two (17), case (17), discrete (17), one (16), mathcal (16), possible (16), analysis (15), density (15), sample (15), may (14), interval (14), theory (14), number (14), but (14), definition (14), its (14), any (14), data (13), also (13), defined (13), same (13), have (13), sets (13), log (12), test (12), model (12), moments (12), isbn (12), subset (12), given (12), cases (12), called (12), sum (11), distributions (10), regression (10), stochastic (10), algebra (10), only (10), where (10), cumulative (10), correlation (9), general (9), transformation (9), they (9), equality (9), sigma (9), outcomes (9), more (9), probabilities (9), coin (9), toggle (8), terms (8), statistical (8), other (8), time (8), vector (8), variance (8), experiment (8), plot (8), range (8), mass (8), mathematical (8), there (8), consider (8), almost (8), equivalence (8), however (8), some (8), left (8), right (8), has (8), infinite (8), often (8), heads (8), dice (8), height (8), wikipedia (7), standard (7), tails (7), chart (7), event (7), these (7), each (7), whose (7), theoretic (7), used (7), particular (7), part (7), absolutely (7), pmf (7), using (6), non (6), process (6), estimator (6), likelihood (6), multivariate (6), linear (6), most (6), theorem (6), average (6), pdf (6), introduction (6), element (6), see (6), even (6), between (6), because (6), their (6), dg_ (6), above (6), borel (6), allows (6), textstyle (6), person (6), represented (6), subsection (6), page (5), randomness (5), index (5), mathematics (5), rank (5), models (5), domain (5), exponential (5), series (5), normal (5), mixed (5), moment (5), line (5), independent (5), article (5), convergence (5), surely (5), stackrel (5), notions (5), typically (5), increasing (5), positive (5), will (5), begin (5), theta (5), end (5), geq (5), colon (5), finite (5), length (5), instead (5), described (5), children (5), table (4), contents (4), search (4), last (4), retrieved (4), control (4), population (4), least (4), estimation (4), box (4), statistic (4), categorical (4), anova (4), bernoulli (4), bayesian (4), power (4), collection (4), normalization (4), transform (4), links (4), new (4), various (4), law (4), main (4), suppose (4), different (4), underlying (4), equivalent (4), practical (4), useful (4), qquad (4), denoted (4), generating (4), inverse (4), change (4), every (4), corresponding (4), both (4), lebesgue (4), countable (4), taken (4), take (4), green (4), otherwise (4), expected (4), how (4), special (4), here (4), when (4), outcome (4), over (4), intervals (4), type (4), would (4), based (4), toss (4), takes (4), subinterval (4), 360 (4), being (4), examples (4), cdots (4), graph (4), hide (4), move (4), sidebar (4), view (3), about (3), under (3), commons (3), articles (3), short (3), portal (3), system (3), design (3), first (3), product (3), limit (3), survival (3), squares (3), conditional (3), cross (3), partial (3), specific (3), tests (3), factor (3), freedom (3), generalized (3), adaptive (3), inference (3), median (3), chi (3), squared (3), unbiased (3), distance (3), family (3), order (3), natural (3), study (3), trial (3), size (3), unit (3), min (3), scatter (3), matrix (3), deviation (3), 2014 (3), press (3), processes (3), modern (3), measures (3), wiley (3), 978 (3), 2020 (3), practice (3), symmetry (3), references (3), certain (3), instance (3), sequences (3), neq (3), since (3), difference (3), hbox (3), ess (3), sup (3), represents (3), infty (3), zero (3), quad (3), total (3), thus (3), properties (3), derivative (3), negative (3), previous (3), find (3), cdf (3), 5pt (3), calculated (3), mathrm (3), let (3), necessarily (3), obtain (3), rightarrow (3), axiomatic (3), countably (3), relation (3), decreasing (3), either (3), mathematically (3), what (3), check (3), measurability (3), generated (3), member (3), quantity (3), quantities (3), singular (3), spinner (3), result (3), formally (3), interest (3), represent (3), roll (3), delta (3), integer (3), elements (3), than (3), sometimes (3), define (3), assigns (3), independence (3), term (3), extensions (3), image (3), events (3), tools (3), languages (2), contact (2), privacy (2), policy (2), wikimedia (2), use (2), was (2), unsourced (2), statements (2), description (2), wikidata (2), category (2), yale (2), united (2), national (2), information (2), science (2), engineering (2), studies (2), clinical (2), applications (2), proportional (2), spectral (2), frequency (2), autoregressive (2), fuller (2), structural (2), seasonal (2), adjustment (2), stationarity (2), decomposition (2), cluster (2), principal (2), contingency (2), degrees (2), poisson (2), binomial (2), nonparametric (2), ordinary (2), simple (2), equations (2), validation (2), coefficient (2), determination (2), pearson (2), maximum (2), posterior (2), bayes (2), way (2), lehmann (2), ratio (2), score (2), parametric (2), uniformly (2), hypotheses (2), bootstrap (2), mean (2), minimum (2), method (2), point (2), optimal (2), location (2), scale (2), shape (2), parameter (2), sampling (2), designs (2), scientific (2), randomized (2), controlled (2), error (2), survey (2), missing (2), component (2), reduction (2), cleaning (2), scaling (2), processing (2), dependence (2), dispersion (2), central (2), variation (2), harmonic (2), geometric (2), arithmetic (2), zukerman (2), moshe (2), 2001 (2), encyclopedia (2), external (2), kallenberg (2), olav (2), 2nd (2), berlin (2), springer (2), verlag (2), 1995 (2), york (2), billingsley (2), patrick (2), fristedt (2), gray (2), 1996 (2), approach (2), literature (2), university (2), oclc (2), 9781466575592 (2), 2005 (2), www (2), mackey (2), george (2), cambridge (2), machine (2), learning (2), inline (2), citations (2), variate (2), senses (2), significant (2), results (2), large (2), expectation (2), rarely (2), explicitly (2), construct (2), considered (2), practically (2), once (2), them (2), notion (2), associated (2), sure (2), need (2), exists (2), identically (2), distributed (2), closed (2), convex (2), degree (2), monotonic (2), formula (2), aligned (2), expression (2), according (2), invertibility (2), found (2), sides (2), respect (2), complex (2), etc (2), itself (2), those (2), determined (2), known (2), small (2), interpretation (2), captured (2), concept (2), could (2), ask (2), forall (2), observation (2), common (2), choice (2), open (2), giving (2), makes (2), easier (2), well (2), preimage (2), follows (2), formal (2), map (2), difficulties (2), introduce (2), mixture (2), like (2), rational (2), flipped (2), spun (2), neither (2), nor (2), everywhere (2), curv (2), depends (2), uniform (2), quantile (2), indicator (2), sim (2), subseteq (2), direction (2), directions (2), north (2), convenient (2), likely (2), 180 (2), say (2), usually (2), describe (2), representing (2), rolled (2), pair (2), maps (2), fair (2), tfrac (2), 6pt (2), questions (2), whether (2), computation (2), individual (2), less (2), reference (2), joint (2), always (2), written (2), words (2), word (2), vocabulary (2), language (2), shows (2), mapped (2), diagram (2), appearance (2), upload (2), file (2), changes (2), history (2), read (2), english (2), bahasa (2), create (2), account (2), donate (2), menu (2), add, topic, mobile, cookie, statement, developers, code, conduct, legal, safety, contacts, disclaimers, available, additional, apply, site, you, agree, registered, trademark, profit, organization, foundation, inc, creative, attribution, sharealike, license, edited, june, 2026, utc, hidden, categories, october, 2018, matches, https, org, php, title, random_variable, oldid, 1358212673, lux, israel, bnf, france, states, gnd, international, authority, databases, wikiproject, kriging, geostatistics, geographic, environmental, cartography, spatial, psychometrics, official, accounts, jurimetrics, econometrics, demography, crime, census, actuarial, social, identification, reliability, quality, probabilistic, methods, chemometrics, medical, epidemiology, trials, bioinformatics, biostatistics, nelson, aalen, hazard, hitting, accelerated, failure, aft, hazards, kaplan, meier, whittle, wavelet, fourier, autoregression, var, heteroskedasticity, arch, arima, jenkins, arma, xcf, pacf, autocorrelation, acf, breusch, godfrey, durbin, watson, ljung, johansen, dickey, granger, causality, break, cointegration, smoothing, trend, elliptical, equation, classification, discriminant, canonical, components, manova, cochran, mantel, haenszel, mcnemar, graphical, cohen, kappa, covariance, partition, regressions, logistic, families, homoscedasticity, heteroscedasticity, robust, isotonic, semiparametric, nonlinear, predictors, template, splines, mars, simultaneous, effects, errors, residuals, confounding, credible, prior, van, der, waerden, ordered, alternative, jonckheere, terpstra, friedman, kruskal, wallis, mann, whitney, hodges, signed, wilcoxon, sign, bic, aic, selection, normality, shapiro, wilk, jarque, bera, lilliefors, anderson, darling, kolmogorov, smirnov, goodness, fit, student, wald, lagrange, multiplier, multiple, comparisons, randomization, permutation, powerful, testing, jackknife, resampling, tolerance, prediction, pivot, confidence, plug, scheffé, rao, blackwellization, estimators, estimating, frequentist, robustness, asymptotics, divergence, efficiency, loss, decision, functional, sufficiency, completeness, monotone, specification, empirical, quasi, sectional, cohort, observational, down, approximation, assignment, interaction, factorial, blocking, experiments, questionnaire, opinion, poll, stratified, methodology, replication, effect, detrending, differencing, preprocessing, dimensionality, truncation, winsorizing, outlier, max, standardization, feature, fisher, anscombe, stabilizing, yeo, johnson, cox, transformations, ecdf, heatmap, violin, stem, leaf, display, run, radar, pie, histogram, forest, fan, correlogram, biplot, bar, graphics, spearman, kendall, grouped, summary, tables, count, skewness, kurtosis, percentile, interquartile, absolute, mode, lehmer, heinz, heronian, cubic, contraharmonic, center, descriptive, outline, basic, topics, 1307, 2968, arxiv, queueing, teletraffic, 1994, ems, 1965, 9th, tokyo, 119981, mcgraw, hill, papoulis, athanasios, 387, 95313, foundations, 1986, 4th, 0854102, 394960, akademie, 8126517719, bert, lawrence, boston, birkhäuser, 7643, 3807, steigerwald, douglas, california, santa, barbara, 2013, april, economics, 245a, 2002, tsitsiklis, john, τσιτσικλής, γιάννης, belmont, athena, 51441829, 188652940x, bertsekas, dimitri, 3rd, 187, castañeda, arunachalam, dharmaraja, 2012, 9781118344941, dekking, frederik, michel, kraaikamp, cornelis, lopuhaä, hendrik, paul, meester, ludolf, erwin, 1431, 875x, issn, 85233, 896, 1007, 84628, 168, doi, texts, stat, edu, yates, daniel, starnes, daren, 2003, archived, original, 7167, 4773, freeman, moore, david, mathsisfun, com, july, 1980, bulletin, american, society, exploitation, historical, blitzstein, joe, hwang, jessica, crc, deisenroth, marc, peter, aldo, faisal, cheng, soon, ong, kingdom, 1104219401, 108, 47004, relationships, among, generator, compact, observable, pairwise, aleatoricism, sequence, converge, explained, theme, consists, obtaining, similarly, therefore, third, covariances, somewhat, subtle, essentially, relate, remain, isolation, characterized, characterizable, finally, sense, essential, supremum, purposes, strong, actual, following, having, provides, checking, laplace, iid, strength, precise, below, several, preserve, negativity, integral, forming, combination, combinations, convolution, noncentral, differently, compute, distinct, 2f_, halves, fixed, composition, true, procedure, allowed, df_, citation, needed, formulas, densities, demand, admits, roots, bigl, bigr, assuming, differentiating, differentiability, invertible, extended, applying, identity, red, blue, constructed, uses, iverson, bracket, nominal, generalised, class, fully, characterise, problem, characterised, parameters, enough, know, far, question, answered, viewed, intuitively, obtained, members, evaluations, generates, suffices, prove, fact, recall, moreover, simply, note, usual, completion, preimages, topological, intuitive, gives, behaved, leading, means, enables, target, looking, assumption, involves, along, arise, insufficiently, constrained, necessary, termed, constrain, normally, derived, directly, intersections, unions, banach, tarski, paradox, generally, concentrated, dense, refinement, preceding, ranges, half, realized, weighted, cdfs, implies, falling, normalized, samples, desired, calculating, exploits, unifying, framework, randomly, support, unitarity, axiom, choose, horizontal, west, east, south, southeast, commonly, done, mapping, bearing, clockwise, parts, equally, angle, selected, assigned, choosing, speaking, multiplying, integrating, correspond, exact, prescribed, lie, admit, characterize, mixes, arbitrarily, never, occurring, gaps, rolling, obvious, representation, pairs, plotted, picture, columns, payoff, successful, bet, taking, enumeration, gets, constant, piecewise, step, suppressed, hard, treated, measured, computed, persons, track, relationship, acknowledged, come, correlated, posed, another, including, adding, pmfs, chosen, interpreted, 190, 150, 200, push, forward, technical, device, guarantee, existence, disposes, altogether, just, puts, whole, works, development, counting, radon, nikodym, dp_, recording, outputs, yields, forgets, records, output, points, provided, field, ldots, vertices, specify, adjacency, times, sentence, serves, into, alternatively, equals, position, indicates, particularly, disciplines, fields, interested, modeling, numerical, nonetheless, optionally, covary, structures, computer, valid, specifically, refer, manifolds, shapes, trees, matrices, vectors, boolean, traditionally, limited, structure, describes, uncountably, must, 399, many, contexts, denote, form, mid, meaningful, needs, belong, capital, roman, letters, triple, think, systematically, pafnuty, chebyshev, important, possibilities, especially, wherein, automatically, realization, consideration, identical, differ, ways, pushforward, informally, fundamental, chance, uncertainty, philosophically, complicated, straightforward, purely, interpretational, upon, rigorous, setup, measurement, die, defining, might, upper, tossing, formalization, object, refers, variability, aleatory, tree, venn, boole, inequality, marginal, complementary, walk, observed, markov, chain, pareto, singleton, mutual, exclusivity, elementary, collectively, exhaustive, indeterminism, determinism, axioms, phenomenon, free, item, projects, printable, version, download, print, export, get, shortened, url, cite, permanent, link, related, actions, talk, tiếng, việt, اردو, українська, türkçe, tagalog, ไทย, తెలుగు, தமிழ், svenska, sunda, српски, srpski, shqip, slovenščina, srpskohrvatski, српскохрватски, sicilianu, русский, română, português, polski, norsk, bokmål, nederlands, melayu, монгол, македонски, latviešu, 한국어, қазақша, ქართული, 日本語, italiano, íslenska, ido, indonesia, հայերեն, magyar, हिन्दी, עברית, galego, gaeilge, français, suomi, فارسی, euskara, eesti, español, esperanto, ελληνικά, deutsch, dansk, cymraeg, чӑвашла, čeština, qırımtatarca, català, বাংলা, български, беларуская, башҡортса, azərbaycanca, asturianu, العربية, afrikaans, top, personal, pages, recent, community, learn, help, contribute, current, navigation, jump, content,
Text of the page (random words):
lmost never take an exact prescribed value c formally c r pr x c 0 textstyle forall c in mathbb r pr x c 0 but there is a positive probability that its value will lie in particular intervals which can be arbitrarily small continuous random variables usually admit probability density functions pdf which characterize their cdf and probability measures such distributions are also called absolutely continuous but some continuous distributions are singular or mixes of an absolutely continuous part and a singular part an example of a continuous random variable would be one based on a spinner that can choose a horizontal direction then the values taken by the random variable are directions we could represent these directions by north west east south southeast etc however it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers this can be done for example by mapping a direction to a bearing in degrees clockwise from north the random variable then takes values which are real numbers from the interval 0 360 with all parts of the range being equally likely in this case x the angle spun any real number has probability zero of being selected but a positive probability can be assigned to any range of values for example the probability of choosing a number in 0 180 is 1 2 instead of speaking of a probability mass function we say that the probability density of x is 1 360 the probability of a subset of 0 360 can be calculated by multiplying the measure of the set by 1 360 in general the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set more formally given any interval i a b x r a x b textstyle i a b x in mathbb r a leq x leq b a random variable x i u i u a b displaystyle x_ i sim operatorname u i operatorname u a b is called a continuous uniform random variable curv if the probability that it takes a value in a subinterval depends only on the length of the subinterval this implies that the probability of x i displaystyle x_ i falling in any subinterval c d a b displaystyle c d subseteq a b is proportional to the length of the subinterval that is if a c d b one has pr x i c d d c b a displaystyle pr left x_ i in c d right frac d c b a where the last equality results from the unitarity axiom of probability the probability density function of a curv x u a b displaystyle x sim operatorname u a b is given by the indicator function of its interval of support normalized by the interval s length f x x 1 b a a x b 0 otherwise displaystyle f_ x x begin cases displaystyle 1 over b a a leq x leq b 0 text otherwise end cases of particular interest is the uniform distribution on the unit interval 0 1 displaystyle 0 1 samples of any desired probability distribution d displaystyle operatorname d can be generated by calculating the quantile function of d displaystyle operatorname d on a randomly generated number distributed uniformly on the unit interval this exploits properties of cumulative distribution functions which are a unifying framework for all random variables mixed type edit a mixed random variable is a random variable whose cumulative distribution function is neither discrete nor everywhere continuous 10 it can be realized as a mixture of a discrete random variable and a continuous random variable in which case the cdf will be the weighted average of the cdfs of the component variables 10 an example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads if the result is tails x 1 otherwise x is the value of the spinner as in the preceding example there is a probability of 1 2 that this random variable will have the value 1 other ranges of values would have half the probabilities of the last example most generally every probability distribution on the real line is a mixture of discrete part singular part and an absolutely continuous part see lebesgue s decomposition theorem refinement the discrete part is concentrated on a countable set but this set may be dense like the set of all rational numbers measure theoretic definition edit the most formal axiomatic definition of a random variable involves measure theory continuous random variables are defined in terms of sets of numbers along with functions that map such sets to probabilities because of various difficulties e g the banach tarski paradox that arise if such sets are insufficiently constrained it is necessary to introduce what is termed a sigma algebra to constrain the possible sets over which probabilities can be defined normally a particular such sigma algebra is used the borel σ algebra which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and or intersections of such intervals 11 the measure theoretic definition is as follows let ω f p displaystyle omega mathcal f p be a probability space and e e displaystyle e mathcal e a measurable space then an e e displaystyle e mathcal e valued random variable is a measurable function x ω e displaystyle x omega to e which means that for every subset b e displaystyle b in mathcal e its preimage is f displaystyle mathcal f measurable x 1 b f displaystyle x 1 b in mathcal f where x 1 b ω x ω b displaystyle x 1 b omega x omega in b 12 this definition enables us to measure any subset b e displaystyle b in mathcal e in the target space by looking at its preimage which by assumption is measurable in more intuitive terms a member of ω displaystyle omega is a possible outcome a member of f displaystyle mathcal f is a measurable subset of possible outcomes the function p displaystyle p gives the probability of each such measurable subset e displaystyle e represents the set of values that the random variable can take such as the set of real numbers and a member of e displaystyle mathcal e is a well behaved measurable subset of e displaystyle e those for which the probability may be determined the random variable is then a function from any outcome to a quantity such that the outcomes leading to any useful subset of quantities for the random variable have a well defined probability when e displaystyle e is a topological space then the most common choice for the σ algebra e displaystyle mathcal e is the borel σ algebra b e displaystyle mathcal b e which is the σ algebra generated by the collection of all open sets in e displaystyle e in such case the e e displaystyle e mathcal e valued random variable is called an e displaystyle e valued random variable moreover when the space e displaystyle e is the real line r displaystyle mathbb r then such a real valued random variable is called simply a random variable note that we are not giving r displaystyle mathbb r the usual lebesgue σ displaystyle sigma algebra which is the completion of the borel σ displaystyle sigma algebra this choice allows for more measurable functions f ω r displaystyle f omega to mathbb r and makes it easier to check that a function f ω r displaystyle f omega to mathbb r is measurable as we only need to check that preimages of open sets are measurable real valued random variables edit in this case the observation space is the set of real numbers recall ω f p displaystyle omega mathcal f p is the probability space for a real observation space the function x ω r displaystyle x colon omega rightarrow mathbb r is a real valued random variable if ω x ω r f r r displaystyle omega x omega leq r in mathcal f qquad forall r in mathbb r this definition is a special case of the above because the set r r r displaystyle infty r r in mathbb r generates the borel σ algebra on the set of real numbers and it suffices to check measurability on any generating set here we can prove measurability on this generating set by using the fact that ω x ω r x 1 r displaystyle omega x omega leq r x 1 infty r moments edit the probability distribution of a random variable is often characterised by a small number of parameters which also have a practical interpretation for example it is often enough to know what its average value is this is captured by the mathematical concept of expected value of a random variable denoted e x displaystyle operatorname e x and also called the first moment in general e f x displaystyle operatorname e f x is not equal to f e x displaystyle f operatorname e x once the average value is known one could then ask how far from this average value the values of x displaystyle x typically are a question that is answered by the variance and standard deviation of a random variable e x displaystyle operatorname e x can be viewed intuitively as an average obtained from an infinite population the members of which are particular evaluations of x displaystyle x mathematically this is known as the generalised problem of moments for a given class of random variables x displaystyle x find a collection f i displaystyle f_ i of functions such that the expectation values e f i x displaystyle operatorname e f_ i x fully characterise the distribution of the random variable x displaystyle x moments can only be defined for real valued functions of random variables or complex valued etc if the random variable is itself real valued then moments of the variable itself can be taken which are equivalent to moments of the identity function f x x displaystyle f x x of the random variable however even for non real valued random variables moments can be taken of real valued functions of those variables for example for a categorical random variable x that can take on the nominal values red blue or green the real valued function x green displaystyle x text green can be constructed this uses the iverson bracket and has the value 1 if x displaystyle x has the value green 0 otherwise then the expected value and other moments of this function can be determined functions of random variables edit a new random variable y displaystyle y can be defined by applying a real borel measurable function g r r displaystyle g colon mathbb r rightarrow mathbb r to the outcomes of a real valued random variable x displaystyle x that is y g x displaystyle y g x the cumulative distribution function of y displaystyle y is then f y y p g x y displaystyle f_ y y operatorname p g x leq y if function g displaystyle g is invertible i e h g 1 displaystyle h g 1 exists where h displaystyle h is g displaystyle g s inverse function and is either increasing or decreasing then the previous relation can be extended to obtain f y y p g x y p x h y f x h y if h g 1 increasing p x h y 1 f x h y if h g 1 decreasing displaystyle f_ y y operatorname p g x leq y begin cases operatorname p x leq h y f_ x h y text if h g 1 text increasing operatorname p x geq h y 1 f_ x h y text if h g 1 text decreasing end cases with the same hypotheses of invertibility of g displaystyle g assuming also differentiability the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to y displaystyle y in order to obtain 10 f y y f x h y d h y d y displaystyle f_ y y f_ x bigl h y bigr left frac dh y dy right if there is no invertibility of g displaystyle g but each y displaystyle y admits at most a countable number of roots i e a finite or countably infinite number of x i displaystyle x_ i such that y g x i displaystyle y g x_ i then the previous relation between the probability density functions can be generalized with f y y i f x g i 1 y d g i 1 y d y displaystyle f_ y y sum _ i f_ x g_ i 1 y left frac dg_ i 1 y dy right where x i g i 1 y displaystyle x_ i g_ i 1 y according to the inverse function theorem the formulas for densities do not demand g displaystyle g to be increasing in the measure theoretic axiomatic approach to probability if a random variable x displaystyle x on ω displaystyle omega and a borel measurable function g r r displaystyle g colon mathbb r rightarrow mathbb r then y g x displaystyle y g x is also a random variable on ω displaystyle omega since the composition of measurable functions is also measurable however this is not necessarily true if g displaystyle g is lebesgue measurable citation needed the same procedure that allowed one to go from a probability space ω p displaystyle omega p to r d f x displaystyle mathbb r df_ x can be used to obtain the distribution of y displaystyle y example 1 edit let x displaystyle x be a real valued continuous random variable and let y x 2 displaystyle y x 2 f y y p x 2 y displaystyle f_ y y operatorname p x 2 leq y if y 0 displaystyle y 0 then p x 2 y 0 displaystyle p x 2 leq y 0 so f y y 0 if y 0 displaystyle f_ y y 0 qquad hbox if quad y 0 if y 0 displaystyle y geq 0 then p x 2 y p x y p y x y displaystyle operatorname p x 2 leq y operatorname p x leq sqrt y operatorname p sqrt y leq x leq sqrt y so f y y f x y f x y if y 0 displaystyle f_ y y f_ x sqrt y f_ x sqrt y qquad hbox if quad y geq 0 example 2 edit suppose x displaystyle x is a random variable with a cumulative distribution f x x p x x 1 1 e x θ displaystyle f_ x x p x leq x frac 1 1 e x theta where θ 0 displaystyle theta 0 is a fixed parameter consider the random variable y l o g 1 e x displaystyle y mathrm log 1 e x then f y y p y y p l o g 1 e x y p x l o g e y 1 displaystyle f_ y y p y leq y p mathrm log 1 e x leq y p x geq mathrm log e y 1 the last expression can be calculated in terms of the cumulative distribution of x displaystyle x so f y y 1 f x log e y 1 1 1 1 e log e y 1 θ 1 1 1 e y 1 θ 1 e y θ displaystyle begin aligned f_ y y 1 f_ x log e y 1 5pt 1 frac 1 1 e log e y 1 theta 5pt 1 frac 1 1 e y 1 theta 5pt 1 e y theta end aligned which is the cumulative distribution function cdf of an exponential distribution example 3 edit suppose x displaystyle x is a random variable with a standard normal distribution whose density is f x x 1 2 π e x 2 2 displaystyle f_ x x frac 1 sqrt 2 pi e x 2 2 consider the random variable y x 2 displaystyle y x 2 we can find the density using the above formula for a change of variables f y y i f x g i 1 y d g i 1 y d y displaystyle f_ y y sum _ i f_ x g_ i 1 y left frac dg_ i 1 y dy right in this case the change is not monotonic because every value of y displaystyle y has two corresponding values of x displaystyle x one positive and negative however because of symmetry both halves will transform identically i e f y y 2 f x g 1 y d g 1 y d y displaystyle f_ y y 2f_ x g 1 y left frac dg 1 y dy right the inverse transformation is x g 1 y y displaystyle x g 1 y sqrt y and its derivative is d g 1 y d y 1 2 y displaystyle frac dg 1 y dy frac 1 2 sqrt y then f y y 2 1 2 π e y 2 1 2 y 1 2 π y e y 2 displaystyle f_ y y 2 frac 1 sqrt 2 pi e y 2 frac 1 2 sqrt y frac 1 sqrt 2 pi y e y 2 this is a chi squared distribution with one degree of freedom example 4 edit suppose x displaystyle x is...
|
