J. Comp. & Math. Sci. Vol.2 (4), 669-673 (2011)
A Probability Model for the Total Number of Out-Migrants from the Rural Area HIMANSHU PANDEY Department of Mathematics and Statistics DDU Gorakhpur University, Gorakhpur, U.P., India ABSTRACT An attempt has been made to develop a probability model for rural out migrants in inflated form. The parameters involved in this model have been estimated by estimation technique. The model has been applied to some real set of demographic data for testing its suitability. The proposed model leads to fresh environment and quality of life of urban area population. Keywords: Probability Model, Risk of migration, Out-migration, Household. AMS Subject Classification: 62-P25.
INTRODUCTION The social and cultural aspects of a society affect and are affected by the dynamic processes of the demographic components, viz. fertility, mortality and migration. Except mortality all other demographic phenomena are subject to the influence of behavioral variables and involve human decision. The cultural aspects like norms, attitudes, motivations and social factors like the basis of social organization, socialization processes and the established industrial patterns are relevant. Migrants play a decisive role in the social and economic developments of their household particularly in developing countries. Most of the findings of the migration studies are mainly based on deterministic approach. The number of migrants per household is essentially a random variable and this
requires stochastic approach for its study. Most of the Macro level migration studies Roshier10, Greenwood3, Lansing, Mullar4, Speare7 have failed to account for the tremendous regional and local heterogeneity that prevails in the spatial economy and its movement patterns. Later on micro-level research on both residential mobility and migration has played a decisive role in the development of theory of migration. Show, Speare et.al.8, Dejong et. al.2, Srivastava9, Ojha & Pandey5, Pandey et. al.. The microlevel studies may be made on community level, village level, household level or individual level itself depending on the need and availability of data. In past few decades, a number of studies have been done. Which deal with the distribution of the number of out-migrants at the household level. The micro-level studies
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
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Himanshu Pandey, J. Comp. & Math. Sci. Vol.2 (4), 669-673 (2011)
may be done on community, village, household, families or individual level itself depending on the need and availability of data. In the study area, there are mainly three types of migrants from a household (i) an adult male aged fifteen years and over who migrate alone at his destination leaving his wife and children behind in the village (ii) an individual who migrate with his wife and children, friends etc. and (iii) an adult male migrant (aged 15 and above) migrates to the place of destination taking dependent migrants provided that the number of dependent migrants has to be equal to or more than the number of male migrants from a household. The three types of migrants considered above differ in their characteristics to give rise to a differential pattern in the socio-cultural characteristics of the household. Migrants of above categories maintained their closed lines with their household in the villages by sending remittances, writing letters and visiting the household at regular intervals of time. Micro level studies have successfully been able to fit the distribution of male migrants aged fifteen and above but failed to describe the distribution of the total number of migrants occurring from a household. In this paper a probability model has been derived to meet out this purpose. This model satisfactorily fits sets of absorbed distribution of the total number of migrants at the household level taken from the rural areas.
THE MODEL Let X denote the total number of migrants from a household. A probability distribution to describe the variation in the total number of migrants from a household is derived under the following assumptions: (i) Migrants from a household occur in cultures (in group) and it may be taken as a rare event. Further the risk of occurring a cluster vary from household to household. Let Y denote the number of cluster to a household and follow Poison distribution i.e. p[y = j ] =
e −θ θ j , j!
j = 0,1, 2 .....
(1)
(ii) Let Z denote the number of migrants to a cluster and follow inflated geometric distribution increased by one. (iii) Let l-β and β be the risk at survey point i.e. P[Z=0] = l-β (2) P[Z=k] = β [(1−α)+ αp], P[Z=k] = αβpqk-1
k=1 k=2, 3, 4 ......
Where q = 1-p Under these three assumptions the distribution of X is derived as below: let X=Z1 + Z2 + ........+ZN
(3)
Where N is the total number of clusters from the household. If g(s) is the probability generating function of Z,'s(i = l,2,...N) then (Bailey1)
αps g (s ) = β s (1 − α ) + 1 − s 2
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
(4)
Himanshu Pandey, J. Comp. & Math. Sci. Vol.2 (4), 669-673 (2011)
Further, if h(s) is probability generating function of N i.e. probability generating function (p.g.f.) of poison distribution.
h (s ) = e
θ (1− s )
Then the probability generating function (p.g.f.) of X i.e. Gx(S) is Gx ( s ) = h(g (s ))
ps −θβα 1− 1− qs −θβ (1−α )(1− s ) =e e
(5)
That is the probability generating function of X is equal to the probability generating function of Polya-Aeppli distribution with θ replaced by θ β (1-α). Equation (5) may also be written as
671
Gx(S)=( α0 + α1 s + ....)(b0 + b1s+b2s2 + ....) = a0b0 + (a1b0 + a0b1)S +(a1b0 + a1b1 + a0b2) S2 + ............ k
Or coefficien
t of S k = ∑ a 1b k = i where i=0
ak = Polya - Aeppli distribution with θ replaced θαβ
bk =
e−θβ (1−a ) (βθ (1 − a))k , k = 0,1,2......... k!
Therefore the probability density function of X can be written as
P[x = k ] = 1 − β + β e − θ , k = 0 k {θ(1 − α )}k P[x = k ] = β e +∑ k! i =l −θ
q1 {θ(1 − α )} k −1 1 αθ p ∑ (k − i ) j−1 j! q j=1 k −1
i
for k=1,2,3................ Setting
αθp
=A q and θ (1 − α ) = B
the probability distribution function (p.d.f.) of X is
P[x = k ] = 1 − β + β e − θ , k = 0 k Bk P[x = k ] = β e −θ +∑ k! i =l
i
∑ qi j=1
B k −1 k −1 A j (k − i )! j−1 j!
ESTIMATION: The proposed probability distribution consists of four parameters β, α, θ and p and three of these viz α,θ and p have been estimated from the observed distribution of migrants by equating zeroth,
oneth cell theoretical frequencies to observe frequencies of zeroth, oneth cell and theoretical mean equal to observed mean i.e. 1 − β + βe −θ =
N0 N
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
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Himanshu Pandey, J. Comp. & Math. Sci. Vol.2 (4), 669-673 (2011)
βθˆe −θ {αˆpˆ + (1 − αˆ )} =
βθˆ1 +
APPLICATION
N1 N
αˆqˆ
The proposed model has been applied to the data, given in table, collected under RDPG - 1978 for its suitability and has been found that X2 are insignificant at 5% and 2% level of significance. The probability distribution derived in this paper may be applied in the real situations to test the suitability of total number of migrants at the household level. It may also be utilized in calculating the various probabilities of migrants connected with the process of migration from a household and predictions to the migration rates in a specified rural population.
= X pˆ
Where N0 = Number of observations in zeroth cell N1= Number of observations in oneth cell N = Total No. of observations X = observed mean of the distribution While the value of β has been taken from Ojha & Pandey5.
TABLE Distribution of Observed and Expected No. of households according to the total number of migrants and type of villages No. of Migrants
Semi Urban Observed Expected
Remote Observed Expected
Growth Center Observed Expected
0
1032
1032.00
871
871.00
972
972.00
1
68
67.68
140
139.95
130
129.76
2
23
20.78
52
43.89
32
35.91
3
6
14.32
15
24.85
25
23.02
4 5
10 8
10.26 7.40
14 11
17-91 11.91
10 10
10.63 10.63
6
7
5.27
10
7.97
5
7
1
6
6.21
5
8
16
16
11.31
17
13.67
Total
1171
1135
1135.00
1208.00
1208.00
X
2
d.f.
13.10 1171.00
7.50
6.856
8.795
3.040
4
5
4
α
0.6793
0.6134
0.5140
β
0 .9302
0-9965
0.9745
θ
0.1365
0-2660
0 .2240
P
0.2961
0.3594
0.3323
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
Himanshu Pandey, J. Comp. & Math. Sci. Vol.2 (4), 669-673 (2011)
REFERENCES 1. Bailey N.J.T. The elements of stochastic process with application to the natural sciences, New York, J.Wiley and Sones (1964). 2. Dejong D.E., R.W. Gardner. Migration decision making, New York, Pergamon Press (1981). 3. Greenwood M.J. A regression analysis of migration to urban areas of less developed countries, the case of India, Journal of Regional Science, 11, pp 253262 (1971) 4. Lansing J. B., E. Muller. The geographical mobility of labour, Ann. Arbor: Survey Research Centre, University of Michigan (1967). 5. Ojha V.P. and H. Pandey. 'A modified probability model for out migration;
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Janasamkhya; Vol. 9, No. 2, pp: 77-81 (1991),. H. Pandey & Jai Kishun. A Probability Model on Rural Out-Migration, Janasamkhya; Vol.14 & 15, pp-49-58 (2010). Speare A. (Jr.). Home Ownership, life cycle stage and residential mobility, Demography, 7, pp. 479-458 (1970). Speare A. (Jr.), s. Goldstein W.H. Frey. Residential Mobility migration and metropolitan change, Cambridge: Ballinger Publishing Company (1975). Singh S.N., K.N.S. Yadava Trend in rural out-migration, Rural Demography, Vol. VIII, No. 1, pp. 53-61 (1981). Friedlander D., R.J. Roshier. A study of internal migration in England and Wales, Part I, Population Studies, 19, pp. 239-279 (1966).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)