DISTRIBUTION MIXTURES

Transcription

1 Application Example 7 DISTRIBUTION MIXTURES One fequently deals with andom vaiables the distibution of which depends on vaious factos. One example is the distibution of atmospheic paametes such as wind speed, baometic pessue, tempeatue, pecipitation intensity etc. The distibution of these quantities vaies with weathe conditions and possibly time of the yea. Fo example, wind speed has a vey diffeent distibution duing inte-stom peiods, in extatopical stoms, and in tonadoes. Anothe example is the weight of vehicles on a highway. The distibution of the weight of a geneic vehicle depends on the type of vehicle (say ca, tuck, o bike), as well as the day (weekend vesus weekday) and hou of the day. The last two factos affect the mixtue of vehicle types in a taffic steam. Fo example, in an uban aea, the faction of tucks is highe duing the night than duing the day. A thid example is the stength of concete, the distibution of which depends on the amount of wate used in the mixtue, the amount and type of aggegates and the cuing conditions, among othe factos. Let X be the vaiable of inteest and denote by Θ a facto that influences the distibution of X. Fo simplicity, suppose that Θ is discete, with possible values θ i, i = 1,...,, and denote by A i the event Θ = θ i. An effective technique to obtain the distibution of X is though conditioning, i.e. one fist detemines the conditional distibution of X unde each A i and assesses the pobabilities P[A i ] and then one finds the cumulative distibution of X by using the Total Pobability Theoem as

2 F X (x) = P[ X x] = P[A i ]P[ X xa i ] (1) = P[A i ]F X A i (x) This method of conditioning can be extended to the case when the distibution of X depends on seveal contolling factos. What one needs in this multi-facto case is the conditional distibution of X unde each combination of the contolling factos and the pobabilities of such combinations. By diffeentiating both sides of Eq. 1 with espect to x, one obtains the coesponding expession fo the pobability density function, which is f X (x) = P[A i ]f X A i (x) () A distibution obtained as the weighted sum of othe distibutions, as in Eqs. 1 and, is called a distibution mixtue. As an example, conside the time D it takes to commute in the moning fom a subub to downtown Boston. Fo given taffic and weathe conditions, espectively T and W, the distibution of D might be lognomal, with mean value m D (T,W) and coefficient of vaiation V D (T,W). The lognomal pobability density function is usually witten in tems of two othe paametes, m ln D and σ ln D, which ae the mean value and vaiance of ln(d). This density has the fom f D (d) = 1 1 (lnd m lnd ) /σ e ln D (3) d π σ ln D

3 In tun, m ln D and σ ln D can be calculated fom m D and σ D = m D VD as m ln D = ln(m D ) 1 ln ( σ D + md) σ ln D = ln(m D ) + ln(σ D + m D ) (4) The eason why the distibution in Eq. 3 is called lognomal is that ln(d) has nomal distibution, with mean value and vaiance given by Eq. 4. Poblem 7.1 Fo a given subub of Boston and a cetain oute, values of the distibution paametes m D (T,W) and V D (T,W) and pobabilities of diffeent (T,W) combinations might be as follows (m D in minutes): W T P[T W] m D (minutes) V D good light good nomal good heavy bad light bad nomal bad heavy = 1.00 Use Eq. to find the maginal pobability density function of D. Plot this density function. Is it a lognomal density? Using the fact that, fo any given W and T, ln(d) has nomal distibution with paametes in Eq. 4 and using tables of the nomal distibution, find the unconditional pobability that D > 60 minutes. 3

4 An impotant featue of the conditioning appoach is that, while the pobability distibution of the mixing vaiable Θ may depend on the application, the conditional distibutions of (X Θ) usually do not and can be obtained fom lage data sets. Fo example, conside the case when X = vehicle weight and let Θ be a discete vaiable with values 1,, and 3 fo tuck, ca, and bike, espectively. Then (X Θ) is the weight of a andomly chosen tuck, ca o bike, depending on Θ. The distibution of these conditional vaiables is athe constant nationwide and may be accuately estimated using extensive suveys fom vaious pats of the county. These conditional distibutions ae schematically illustated in Figue 1a. On the othe hand, the taffic mixtue (the distibution of the discete vaiable Θ) vaies fom location to location and must be detemined locally (Figue 1b). Doing so is a much easie task than having to detemine the distibution of X exclusively fom local data. Obtaining the distibution of X fom Eq. 1 makes best use of all available data, at both the local and national levels. 4

5 Figue 1: Illustation of distibution mixtue fo vehicle weight 5