The Minimum of n Independent Normal Distributions

Transcription

1 The Minimum of n Indeendent Normal Distributions Joshua E. josh-math@untruth.org Initial Release: Aril, Current Revision: August, Problem: We have a race of n runners. What is the robability that a articular runner will win? Each runner s event time can be viewed as a random variable, which we ll assume is distributed normally with a runner-secific mean and standard deviation for their time. The j th runner s time is distributed as X j NormalDistribution j ; j (that is, normally distributed with mean j and standard deviation j ). First, let s examine the n D case, that is the chance in a two erson race that the first runner wins. We want Pr.X 0 < X / where X 0 and X are distributed via different normal distributions (assumed indeendent). This is equivalent to Pr..X 0 X / < 0/, so you re subtracting two normal distributions (or adding X 0 with X, it doesn t matter) which results in another normal distribution..x 0 X / NormalDistribution. 0 ; 0 C /, so you get a simle calculation (an integral conducted via numerical methods or a table looku) to figure what Pr..X 0 X / < 0/ is. For more than two runners, it becomes a bit harder. The way one would actually, in ractice, solve this roblem is to rogram a simulation and run many thousands of rounds of simulation given your articular data. For a roer mathematical solution, the robability for layer to win in a race with n C runners total is Pr.X 0 < Y /, where Y D min j n X j That is, Y is the distribution of the smallest time from the remaining n runners. To find the distribution of Y, we first look for its cumulative distribution function (CDF). In the standard setting, we would follow this

2 u by taking the derivative of the CDF to get the PDF, but we don t roceed in this way. We start with some results for a single variable; we ll write the normal comlementary cumulative distribution function as F c. Pr.X j > a/ D F c.a/ D Pr.X j a/ D C erf x j j!! D erf x j j! D erfc a j j!! Where erfc is the comlementary error function, defined as erfc.x/ D erf.x/ D x 0 e t dt D e t dt The robability Y a can be more easily viewed as the robability of the comlement; that is, what is the chance that every X j.j /, is strictly greater than a. This will be in terms of the comlementary cumulative distribution function for the distribution X j, which we ll call F c;j. We can define the cumulative distribution function for Y : Pr.Y a/ D Pr.X > a and X > a and : : : and X n > a/ D Pr.X j > a/ (the distributions are indeendent) D D D D j D F c;j.a/ j D j D j D j D!! erfc a j j j e t a j j a dt e t j j dt D Q n j D n j a e a x P n j D tj j j dt : : : dt n

3 From this last form, we can hrase this in terms of a multivariate normal distribution, with a covariance matrix 0 S D i ı i;j D : :: i;j n 0 n With this convention, taking t t D : and D t n : n Pr.Y a/ D Q n j D n j a D n det S D f.t/ dt.a;/ n e.a;/ n e a P n j D tj j j.t /S.t / T dt dt : : : dt n where f.t/ is the PDF for this multivariate normal distribution f.t/ D det S n e.t /S.t / T We actually need Pr.Y > a/, which, by the above, is R n.a;/ f.t/ dt We ll refer to the PDF of X 0 as g.s/ D want to calculate Pr.X 0 < Y / D D D e 0 Pr.X 0 D s/ Pr.Y > s/ ds g.s/ f.t/ dt ds.s;/ n g.s/f.t/ dt ds.s;/ n.;/.s 0 / 0. We

4 As may be clear from the above, multilying the PDFs from indeendent normal distributions gives a multivariate normal distribution, so the roduct of f and g yields another multivariate distribution in one additional variable; the covariance matrix would be W D i ı i;j D i;j 0 0 : :: n 0 n With this convention, taking v0 0 v D v : and f D : v n n and we get the PDF for this new distribution: h.v/ D det W nc e.v f /W.v f / T so our final robability is: P r.x 0 < Y / D. ;/.s;/ n h.s; t ; : : : ; t n / dtds That s a wonderful clear statement concetually, but doing calculations using this multivariate case is imractical. To do calculations, it is much referable to sto much much sooner in the calculation: Pr.X 0 < Y / D D D Pr.X 0 D s/ Pr.Y > s/ ds g.s/ F c;j.s/ ds j D 0 e.s 0 / 0 j D!! erfc s j ds j The Mathematica code that does this calculation is rovided in Listing.

5 Listing : Direct Calculation Aroach WinProb[layerList_] := Module[{u, v}, u = PDF[layerList[[]], x]; v = Product[Integrate[PDF[layerList[[j]], t], {t, x, Infinity}], {j,, Length[layerList]}]; NIntegrate[u*v, {x, -Infinity, Infinity}] ] Here, we simly ass in a list containing the layer time distribution (the analysis above only suorts assing in normal distributions!). To confirm that this works correctly, we can also create a routine to do this in simulation, rovided in Listing. Listing : Simulation Aroach WinProbSim[layerList_, rounds_] := Module[{wins = 0, cntr, curround}, For[cntr = 0, cntr < rounds, cntr++, curround = Ma[RandomReal, layerlist]; If[curRound[[]] == Min[curRound], wins++] ]; wins/rounds ] Running this roceeds in the same way, other than the fact that we need to tell the routine how many rounds to use in the simulation. The intuitive case (where all runners have exactly the same characteristics) work out as exected (in the n layer case, layer has a robability of of winning. n If we instead examine a race between runners (layer to ) in the meter dash, where all layers have an even time standard deviation of second (wildly high, of course). Runners through have an average event time of. seconds (the world record for this event). Runner s mean time is shown as an advantage from the central time of. seconds, so his mean race time varies from. (advantage is ) to. (advantage is ). The code that imlements this race is resented in Listing. Listing : Race (Calculated) FirstRace = Table[NormalDistribution[9.58, ], {i,, 9}]; AdvData = Table[{adv, WinProb[Preend[FirstRace, NormalDistribution[ adv, ]]]}, {adv, 0, 5,.05}];

6 To run the same race in simulation, we instead use the code in Listing. Listing : Race (Simulation) AdvSimData = Table[{adv, WinProbSim[Preend[FirstRace, NormalDistribution[ adv, ]], 0000]}, adv, 0, 5,.05}]; When the simulation is run with, simulation rounds, the symbolic calculation is very close to the simulated result. Figure deicts these results, with runner s advantage shown on the x-axis and the robability that runner wins shown on the y-axis. Figure : Race Results Let s now examine the situation where there is a wider array of runners. In this race, we ll include runners, with all runners having an event time standard deviation of. seconds (still ridiculously high). Runners - have a mean time of. u to. seconds (they are each searated by / of a second). Runner s mean time is again shown as an advantage from the central time of. seconds, so it varies from. seconds (advantage is ) to. seconds (advantage is ). This advantage is shown on the x-axis. The robability that runner wins is shown on the y-axis. Direct comutation (via numerical aroximation) is accomlished using the code in Listing.

7 Listing : Race (Calculated) SecondRace = Table[NormalDistribution[ offset,.5], {offset, -,, 4/9}]; SecondRaceAdvData = Table[{adv, WinProb[Preend[SecondRace, NormalDistribution[ adv, 0.5]]]}, {adv, 0, 5,.05}]; Simulation is accomlished using the code in Listing. Listing : Race (Simulation) SecondRaceAdvSymData = Table[{adv, WinProbSim[Preend[SecondRace, NormalDistribution[ adv, 0.5]], 0000]}, {adv, 0, 5,.05}]; Again, these roduce very similar results, as seen in Figure. Figure : Race Results

8 E Colohon The text of this document is tyeset in Jean-François Porchez s wonderful Sabon Next tyeface. Sabon Next is a modern ( ) revival of Jan Tschichold s Sabon tyeface, which is in turn a adatation of the classical (in all meanings) Garamond tyeface, which dates from the early th century. Equations are tyeset using the MathTime Professional II (MTPro ) fonts, a font ackage released in by the great mathematical exositor Michael Sivak. These fonts are designed to work with the Times tyeface, but they blend well with most classical fonts. Source listings are tyeset in Microso s Consolas, a monosace font with excellent readability. X TEX was used to tyeset the document, which is in turn offsring of Donald Knuth s rofoundly imortant TEX. X TEX was selected in order to gain access to modern fonts without the trauma involved in converting them to a reresentation that dftex could deal with. This aroach makes most (though sadly, not all) OenTye features available, and sidestes the traditional limit of glyhs er font. WinEdt was used as an editor. Diagrams were roduced in Mathematica. E