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That,,is,,,f,,is,,any,,measurable,,function,,with,,the,,property,,that:.,,U,,,,=,,,,Y,,,,Z,,,,{displaystyle,,,,U=YZ},,,,V,,,,=,,,,Z,,,,{displaystyle,,,,V=Z},,,,.,,,,The,,above,,formulas,,can,,be,,generalized,,to,,variables,,(which,,we,,will,,again,,call,,y),,depending,,on,,more,,than,,one,,other,,variable.,,If,,,the,,,function,,,g,,,is,,,monotonic,,,,then,,,the,,,resulting,,,density,,,function,,,is.,,,See,,,Law,,,of,,,the,,,unconscious,,,statistician.,,,{displaystyle,,,f(x)={frac,,,{1}{sqrt,,,{2pi,,,}}};e^{ -x^{2}/2}.},,,.,,,f,,,X,,,(,,,x,,,),,,=,,,d,,,d,,,x,,,F,,,X,,,(,,,x,,,),,,.,,,E,,,⁡,,,[,,,X,,,],,,=,,,∫,,,−,,,∞,,,∞,,,x,,,f,,,(,,,x,,,),,,d,,,x,,,.,,,p,,(,,Y,,),,=,,∫,,−,,∞,,∞,,p,,U,,(,,Y,,Z,,),,p,,V,,(,,Z,,),,,,Z,,,,d,,Z,,{displaystyle,,p(Y)=int,,{ -infty,,}^{infty,,}p{U}(YZ),p{V}(Z),Z,dZ},,.,,

Then,,,the,,joint,,density,,p(Y,Z),,can,,be,,computed,,by,,a,,change,,of,,variables,,from,,U,V,,to,,Y,Z,,,and,,Y,,can,,be,,derived,,by,,marginalizing,,out,,Z,,from,,the,,joint,,density.,,The,,,,integral,,,,of,,,,f,,,,over,,,,any,,,,window,,,,of,,,,time,,,,(not,,,,only,,,,infinitesimal,,,,windows,,,,but,,,,also,,,,large,,,,windows),,,,is,,,,the,,,,probability,,,,that,,,,the,,,,bacterium,,,,dies,,,,in,,,,that,,,,window.,,,,The,,,modern,,,measure-theoretic,,,foundation,,,of,,,probability,,,theory;,,,the,,,original,,,German,,,version,,,(Grundbegriffe,,,der,,,Wahrscheinlichkeitsrechnung),,,appeared,,,in,,,1933.,,,{displaystyle,,,,f={frac,,,,{dX{*}P}{dmu,,,,}}.},,,,.,,,,,,,f,,,Y,,,(,,,y,,,),,,d,,,y,,,,,,=,,,,,,f,,,X,,,(,,,x,,,),,,d,,,x,,,,,,,,,,{displaystyle,,,leftf{Y}(y),dyright=leftf{X}(x),dxright,},,,.,,,

or.,,,..,,,,for,,,,any,,,,measurable,,,,set,,,,A,,,,∈,,,,A,,,,{displaystyle,,,,Ain,,,,{mathcal,,,,{A}}},,,,.,,,,If,,,,dt,,,,is,,,,an,,,,infinitely,,,,small,,,,number,,,,,the,,,,probability,,,,that,,,,X,,,,is,,,,included,,,,within,,,,the,,,,interval,,,,(t,t dt),,,,is,,,,equal,,,,to,,,,f(t)dt,,,,,or:.,,,,with,,the,,differential,,regarded,,as,,the,,Jacobian,,of,,the,,inverse,,of,,H,,,evaluated,,at,,y.,,(1972),,Families,,of,,Frequency,,Distributions,,,Griffin.,,Furthermore,,,,when,,,it,,,does,,,exist,,,,the,,,density,,,is,,,almost,,,everywhere,,,unique.,,,Hence,,,,,if,,,,FX,,,,is,,,,the,,,,cumulative,,,,distribution,,,,function,,,,of,,,,X,,,,,then:.,,,,

Discussion[edit].,,,,one,,,can't,,,choose,,,the,,,counting,,,measure,,,as,,,a,,,reference,,,for,,,a,,,continuous,,,random,,,variable).,,,f(x1,,,,,,,,,,xn),,,,shall,,,,denote,,,,the,,,,probability,,,,density,,,,function,,,,of,,,,the,,,,variables,,,,that,,,,y,,,,depends,,,,on,,,,,and,,,,the,,,,dependence,,,,shall,,,,be,,,,y,,,,=,,,,g(x1,,,,,,,,,,xn).,,,,This,,,is,,,similarly,,,the,,,case,,,for,,,the,,,sum,,,U V,,,,difference,,,U-V,,,and,,,product,,,UV.,,,This,,alternate,,definition,,is,,the,,following:.,,Dependent,,,,variables,,,,and,,,,change,,,,of,,,,variables[edit].,,,,f,,,,(,,,,x,,,,),,,,=,,,,1,,,,2,,,,π,,,,e,,,,−,,,,x,,,,2,,,,/,,,,2,,,,.,,,,Note,,,,that,,,,it,,,,is,,,,not,,,,possible,,,,to,,,,define,,,,a,,,,density,,,,with,,,,reference,,,,to,,,,an,,,,arbitrary,,,,measure,,,,(e.g.,,,,It,,,is,,,not,,,necessary,,,that,,,g,,,be,,,a,,,one-to-one,,,function.,,,

That,,,,is,.,,,,First,,,the,,variables,,have,,the,,following,,density,,functions:.,,Ushakov,,,N.G.,,Pr,,,,[,,,,a,,,,≤,,,,X,,,,≤,,,,b,,,,],,,,=,,,,∫,,,,a,,,,b,,,,f,,,,X,,,,(,,,,x,,,,),,,,d,,,,x,,,,.,,,,^,,"AP,,Statistics,,Review,,-,,Density,,Curves,,and,,the,,Normal,,Distributions".,,A,,,lot,,,of,,,bacteria,,,live,,,for,,,approximately,,,5,,,hours,,,,but,,,there,,,is,,,no,,,chance,,,that,,,any,,,given,,,bacterium,,,dies,,,at,,,exactly,,,5.0000000000.,,, 4bc16de163

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