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A Probability model for the risk of vulnerability to HIV/AIDS infection among female migrants

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Author : HIMANSHU PANDEY and RAJENDRA TIWARI Published in : Journal of Computer and Mathematical Sciences (http://compmath-journal.org) ABSTRACT The main objective of this paper is to developed an inflated probability model for described and analysis, how the female migrant are more vulnerable to HIV/AIDS. The suitability of the model is tested through observed data.
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  A Probability model for the risk of vulnerabilityto HIV/AIDS infection among female migrants HIMANSHU PANDEY and RAJENDRA TIWARI Department of Mathematics, and Statistics, D.D.U.Gorakhpur University, Gorakhpur (India) ABSTRACT  The main objective of this paper is to developed an inflatedprobability model for described and analysis, how the femalemigrant are more vulnerable to HIV/AIDS. The suitability of themodel is tested through observed data. Key Words:  Inflated Probability Model, DisplacedGeometric Distribution, Method of Moments, MLE.  J. Comp. & Math. Sci. Vol. 1(2), 145-154 (2010). INTRODUCTION Women are working in almostall types of jobs, such as technical,professional and non-professional inboth private and public sectors. So, thetraditional role of women as housewives has gradually changed into workingwomen and housewives (Reddy, 15 ;Anand 2 ). They have also started activelyparticipating in the socio-economicdevelopment of the country. They areworking in almost all types of jobs eitherthat are in Public or Private Sectors. Today,in an increasingly globalized economy,migration often provides an employmentopportunities giving rise to an unpre-cedented flow of migrants, includingincreasing numbers of female migrants(Jhingarn; Bhatt; Desai) 12 . The reasonfor migration is recognized that womenmore within countries in response tothe inequitable distribution of resources, services and opportunities.Migration, especially in the process of regional economic development,urbanization and industrialization is animportant cause and the effect of social and economic change. The socio-cultural characteristics of the householdsare more likely to be affected by femaleand children migration whereas, theeconomic level is affected by the malemigrants. Thus, it is important toinvestigate the variation in the numberof migrants from a household underthis consideration. MODEL: A probability model for thenumber of closed boy friends to describethe distribution of single unmarried  Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)  146 Himanshu Pandey et al. , J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010).female migrants proposed under thefollowing assumption:(i)Let   be the proportion of femalemigrants having at least one closeboy friend.(ii)Out of   proportion of femalemigrants, let   be the proportionof female migrants having only oneclosed boy friends.(iii)Number of close boy friends attachedwith female migrants follows atruncated displaced Geometricdistribution.(iv)Let p be the probability of closeboy friends attached with youngunmarried female migrants, they aremore vulnerable to HIV/AIDS infec-tion.Let the random variable x denotesthe number of closed boy friends.From the above assumptions, theprobability model is given by      1]0[  X P  ; K=0      ]1[  X P  ; K=1    N K  q pqK  X P   1)1(][ 2     ; K=2, 3,……..N (2.1.) The above probability modelinvolves three parameters  ,  , p to beestimated from the observed distributionof female migrants. ESTIMATION: Let N be a known quantity. If Nis taken to be known then the proposedmodel (4.2.1) involves three para-meters  ,    and p only. METHOD OF MOMENT:  The parameters  ,    and p areestimated by equating zeroth and firstcell theoretical frequencies to theobserved frequencies of the respectivecells and theoretical mean equal toobserved mean as follows:    f  f  0 1      (3.1)    f  f  1    (3.2)   q Nqq pq  N  N  N  N    1}1{ 1)1( 11      X q  N    11 (3.3)Where   0  f  =Number of Observed zeroth  Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)   Himanshu Pandey et al. , J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010). 147cell, 1  f  =Number of Observed firstcell,    f  = Total number of observations..and X =Observed mean of the distri-bution. The expected frequencies of thecorresponding cells are obtained aftergetting the estimated values of theparameters by using the above expre-ssions (3.1), (3.2) and (3.3). METHOD OF MAXIMUM LIKELIHOOD: Let x be a random variable froma sample of  f   observation with theprobability function (2.1) where   0  f  denote the number of observation inzenoth cell,   1  f  denote the number of observation in first cell and  f   denotethe total number of observations. Thenthe likelihood function for the givensample can be expressed as:   210 1)1()()1(  f  N  f  f  q p L          210 1)1(  f  f  f  f   N  q p         (3.4)Expression for logarithm of likelihoodfunction is.     0 1 log log 1 log  L f f          2 1log1  N   p f q             012 log  f f f f             11  N   pq         (3.5)Partially differentiating (3.5) with respectto  ,   and p respectively and equatingto zero. We get the following equations.        1 210  f  f  f  LogL     )( 210  f  f  f  f         00 1  f  f  f      0  (3.6)   12 log1  f f  L            012 1  f f f f        Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)      011 1  f f f  f          0   (3.7)   2 log  f  L p p      012 1  N   f f f f q p         0  (3.8)After solving the equations (3.6), (3.7)and (3.8), we get the following estima-ting equations.  0  f f  f      (3.9)  10  f  f f      (3.10)And 201 1  N   f  pq f f f     (3.11) The asymptotic variance of (  ,  , p ) is obtained by investing the infor-mation matrix whose elements arenegatives of second order of thelikelihood function. The second order derivations of  log L  follows from equations (3.6), (3.7)and (3.8) respectively.     2022 log1  f  L          02  f f       (3.12)      2011222 log1  f f f  f  L                (3.13)And  2222  p f  LogL p       2210 )1()(  pq f  f  f  f   N  (3.14)Now 2 log  L        2 log0  L      (3.15)   22 loglog0  L L p p           (3.16)148 Himanshu Pandey et al. , J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010).  Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

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