International Journal of Statistika and Mathematika, ISSN: 2277- 2790 E-ISSN: 2249-8605, Volume 9, Issue 3, 2014 pp 82-84 International Journal of Statistiika and Mathematika, ISSN: 2277- 2790 E-ISSN: 2249-8605, Volume 9 Issue 3 Page 82 Bayesian Analysis of Continuous Fertility Model Himanshu Pandey * , Arun Kumar Rao, Ashivani Kumar Yadav # Department of Mathematics and Statistics, D.D.U. Gorakhpur University, Gorakhpur, Uttar Pradesh, INDIA. Corresponding Addresses: * himanshu_pandey62@yahoo.com , # akyadavballia@gmail.com Research Article Abstract: The present paper aims at exploring a probability model of continuous fertility and also studies its Bayesian analysis under the precautionary loss function. Keyword: Probability Model, Waiting Time, Conception, precautionary Loss Function. Introduction The study of women fecundability has controversially been adopted by different workers. The utility of the study depends upon the proper adoption of the fecundability of women. In most of the available literature, it is found that the fecaundability is assumed to be constant for all women [Singh (1964a), Pathak (1978)]. But in real life there are ample evidences that women vary in their fecundability. So, the fecundability may be thought of as a random variable {Henry (1995), Singh (1964)}. The present work deals with the same concept. Let us suppose that fecundability, say , follows distribution with p.d.f. g (). If T is waiting time for first conception can be treated as random variable which follows the distribution with p.d.f. f(x/) is regarded as a conditional p.d.f. of X for given where marginal probability density function of is given by g() the study can be continued. The Continuous Fertility Model The geometric distribution is being considered as a discrete model for the waiting time first conception as developed by Gini (1924) the continuous model for the analysis of waiting time of first conception. The intuitive properties of exponential distribution also helped in such considerations. For such analysis Geometric distribution was replaced by the exponential distribution. Thus if x denotes the time of first conception, then its probability density function, say f(x;) is given by f (x; ) = e ; x > 0, > 0 (2.1.1) Where is instantaneous fecundability. The survival function, say S (x) is given by S (x) = P[X > x] = e dx = Or S(x)=e (2.1.2) And the conception rate, say w(x) will be W(x) = () () = = (2.1.3) Maximum Likelihood Estimator F (x ∕) = f(x ; ) = e (2.1.4) Where z = x Bayesian Analysis of the Model The first conception of the family is also a part of the past family back ground; therefore Bayesian analysis of conception seems realistic on the basis of some history. In the some coming section the Bayesian analysis has been done for a continuous time model i.e. exponential distribution. We have f(x/) = e ; x > 0, > 0 Where is the instantaneous fecundibility. The fundamental problems in Bayesian Analysis are that of the choice of prior distribution g() and a loss function L ( , ). Let us consider three prior distribution of to obtain the Bayes estimators which are as follows: (i) Quasi-Prior For the situation where the experimenter has no prior information about the parameter , one may use the quasi density ass given by g () = " ; > 0, d > 0 (2.1.5) Here d = 0 leads to a diffuse prior and d = 1, a non informative prior. (ii) Natural Conjugate Prior of The most widely used prior distribution of is the inverted gamma distribution with parameters and (>0) with p.d.f. given by g $ () = % () (&) e ; > 0 (, )>0 0 ; otherwise (2.1.6) The main reason for general acceptability is the mathematical tractability resulting from the fact that inverted gamma distribution is conjugate prior for . (iii) Uniform Prior