**Introduction**

In the theory of reliability and life testing analysis the exponential distribution has a great importance. The main features of the utility of the distribution are its positive ranges its property of forgetfulness and the graphic shape of the distribution. It has a mirror shape *J* when plotted on the range of random variable x , at Χ-axis with respect to its probability density function at Υ-axis. The shape of the distribution supports the idea of life that is in the beginning when the product is recently created the probability of its life is highest and as the time (the random variable x ) increase the probability of its life decreases.

Exponential distribution is second commonly used distribution in the theory of Statistics after the Normal distribution but it is first in the applications of reliability and life testing analysis, it is used more than the normal distribution.

A modification of the well-known exponential life testing model:

when the parameter Θ is itself a random variable distributed uniformly over a finite range has recently been considered by Bhattacharya and Holla (1965). An interpretation of this uniform distribution in terms of a prior density for the parameter will be of direct interest for a Bayesian analysis. In the sequel, the term Bayesian refers to any use or user of prior densities on a parameter space with the associated application of Bayes’s theorem in the analysis of a statistical problem. Such an analysis rests on the belief that in most practical situations, the mathematician will possess some subjective a priori information concerning the probable values of the parameter.

This information may often be reasonably summarized and made objective by the choice of a suitable prior distribution on the parameter space. Another reason that is usually advanced in support of the use of prior densities is the ‘admissibility’ of the Bayes estimates as discussed by Karlin and Rubin (1956) under the general set up of densities with monotone likelihood ratio, of which (1) is, of course, a member. The object of the present paper is to exploit the use of suitable prior information in the framework of life testing techniques that are currently used in practice. A systematic development of the subject of life testing originated from the work of Epstein and Sobel (1953), and the subsequent progress made in the field can be gauged from the two workers, Mendenhall (1958) and Govindarajulu (1964). To date, the most commonly employed procedures are those in which a life test is terminated either after a pre assigned number of failures have taken place or at a pre assigned time point, and the statistical inferences are based on a statistic representing the accumulated life.

**Prior Distributions**

These situations have been examined in detail here in the light of prior information about the parameter Θ, that is available to the life tester from his past experience. It frequently happens that the experimenter knows in advance that the probable values of the parameter lie over a finite range [α,β] but he does not have any strong prior opinions about any subject of values over this range. In such a case, a uniform distribution of over the range mentioned above my serve as a nice approximation to the prior density representing the experimenter’s beliefs. More generally, we shall consider a prior density:

For the case α=0, equation (2) reduces to a uniform density on [α,β] . The reasons for considering the general form (2) will be apparent in due course. The problem will be examined for two other prior densities, namely the exponential density:

and the inverted gamma density of Ralffa and Schalfer [16]

By suitably adjusting the parameters involved, one may be able to reasonably express the decision-makers prior judgments in terms of these prior densities. An excellent account of the choice of prior densities on the parameter space that are capable of summarizing experimenter’s prior beliefs has been given by Raiffa and Schlaifer(1961) and may be consulted for the details. The prior density (3) for the parameter involved in (1) gives rise to a Bessel-function model in life testing and has recently been discussed by Bhattacharya (1967). The reasons for considering the inverted gamma prior density (4) are that it is flexible enough to capture almost any kind of prior experience, and it also possesses the attractive property that the posterior distribution of the parameter after the sample has been observed is also of the inverted gamma type. A family of prior densities which gives rise to posteriors belonging to the same family is very useful in as much as the mathematical tractability is maintained, and this ‘nice’ property has been termed ‘closure under sampling’ by Wetherill(1961). For densities which admit sufficient statistics of fixed dimensionality, Raiffa and Schlaifer(1961) have considered a method of generating prior densities on the parameter space that possess this desirable property. A family of such densities has been called by them a ‘natural conjugate family’, and for the exponential density (1), the inverted gamma prior forms such a family. Under the assumption of the prior densities mentioned above, reliability estimation will also be considered by using the Bayesian approach and a situation in which the times to failure of the test items are not available Ehrenfeld (1962), Bhattacharya (1962) will be discussed. A difficulty arises when the prior information available to the experimenter is rather ‘vague’, or worse still, there is no prior information whatever. What does a Bayesian life tester do in such a situation? Can he objectively assign prior densities on the parameter space in such a state of ‘ignorance’? This has led to the consideration of what are known as ‘prior quasi-densities’ which assign infinite measures on the parameter space. The use and justification of such densities in the framework of life testing problem will be discussed in some detail, and it will be noticed that a prior quasi-density of the form 1/θ^{2} leads to an estimate which identifies with the classical minimum variance un–biased (MVU) estimate of Θ, Epstein and sobel(1953) based on the first ordered observations. The classical result of Pugh(1963) for the MVU estimate of the reliability function will be shown to closely resemble the Bayes estimate obtained under the assumption of a uniform prior Θ for over the entire positive real line.

**Bayesian Estimation of Reliability Function When Uniform Distribution is Taken as a Prior Distribution**

In this section we shall consider the case when the prior density of involved in (1) is restricted to a finite range [α,β] as specified by the family (2). We assume that items from the population (1) are placed on a life test and the experiment is continued till the first * r* failures are observed x_{1},x_{2},x_{3}…..,x_{r} If denote the first *r* ordered observations, then the sample likelihood conditional on Θ is given by:

where

Under the assumption of the prior density (2), the posterior distribution of after observed the first ordered values in a sample of size , can be obtained by using Bayes’s theorem as;:

which is a truncated ‘inverted gamma density’.

The Bayes estimate of the mean life is, therefore, provided by the posterior mean:

where

defines an incomplete gamma function for n>0, the condition being tacitly assumed in the sequel. The variance of the estimate obtained at (8) works out to be:

Where, for brevity, we write

For the case in which the complete sample (x_{1},x_{2},x_{3}…..x_{n}). is available, the results follow from the preceding ones by a mere substitution *r*=n , and T_{r} then reduces to the sample total *nx* . If, however, a life test is terminated at a pre-assigned time point t , the number of failures during that period would be a random variable. If *r* be the observed value in a sample and if the corresponding life-times be x_{1},x_{2},x_{3}…..x_{r}_{ }, then the conditional likelihood (for a given Θ ) of such a sample has a kernel

where,

Under the assumption of a prior density (2), the posterior distribution of Θ turns out to be the same as (7) except that T_{r} has now to be replaced by T’_{r}. Thus, the results for a life test terminated at a pre-assigned time point follow from the preceding ones and in the sequel therefore, we shall restrict our–selves to the case of first ordered observations only. Note that in either case the statistic (T_{r }or T’_{r }) denotes the accumulated life at the termination of the experiment.

A mere substitution α=0 in the expressions given above leads to corresponding results for the uniform prior density over [α,β] to which we now confine our attention, and obtain the Bayesian estimate of the reliability function

Reliability problems have currently drawn the attention of various research workers in the field and the classical minimum variance unbiased (MVU) estimate of the reliability function for an exponential distribution has been obtained by Pugh(1963) based on a complete sample. The Bayesian method considered here applies to a life test that is terminated after the first *r* failures out of a sample of n items. Under the assumption of a uniform prior over [α,β] , the prior distribution of the reliability function at (14) is given by a probability density:

However, when a life test is terminated and the statistic T_{r} is observed, the posterior distribution of Θ computed at (7) for α=0 is used to obtain the posterior distribution of the reliability function. Thus, for values of *Z* in the range stated at (15), the cdf of the posterior distribution of the reliability function is given by:

Now, a differentiation with respect to z yields the probability density:

The posterior mean of this distribution provides the Bayes’ estimate of reliability as:

and the variance of this estimate works out to be:

These results will be used later to obtain estimates for situations when the prior knowledge about *Θ* is slight or none whatever. Such results provide interesting comparisons with corresponding results of the classical theory.

**Bayesian Estimation of Reliability Function when Inverted Gamma Distribution is Taken as a Prior Distribution**

First the ‘natural conjugate family’ of prior densities has been considered for the parameter . This is an inverted gamma family as given by (4). Under this assumption, one can use the sample likelihood conditional on Θ stated at (5) to obtain the posterior distribution:

which, is again an inverted gamma density with parameters (μ+T_{r}) (*r+v *)and thus, the inverted gamma prior distribution is ‘closed under sampling’. The Bayes estimate of the mean life and its variance are, therefore, provided by:

As regards the reliability estimation, the posterior distribution of given at (20) is used to obtain the Bayes estimate.

and the variance of this estimate works out as:

When the prior distribution of *Θ *is given by an exponential density (3), we obtain, by virtue of Bayes’s theorem applied to (5), the posterior distribution:

The denominator in the last expression is evaluated by using an integral representation of *K *(*z*) , the modified Bessel function of the third kind of order *v* , Erdelyi et al. (1953) and subsequent use of the same formula in conjunction with the fact that:

yields for the Bayes estimate:

The Variance of this estimate is evaluated as;:

Based on the posterior distribution (25), The Bayes estimate of the reliability function is obtained as:

and the variance of this estimate comes out as:

**Bayesian Estimates Without the Information of Prior Distribution**

In this section the situation where the experimenter does not have any prior knowledge about the parameter has been considered. The problem confronting a Bayesian life tester is then the choice of suitable prior densities on the parameter space. In such a case neither can he make any prior guess about the parameter values lying over a finite subset of the infinite parameter space nor can he straightaway proceed to choose a suitable prior from the ‘natural conjugate’ family or some other family of densities. One customary way out of the dilemma is to use a constant density *g*(*Θ*)=1 over the positive real line 0<*Θ*< ∞. The uniform prior over the entire real line has frequently been used by Jeffrey (1961) and has also been a subject of much controversy amongst the Bayesian theoreticians for various reasons besides the one that it cannot be interpreted as a probability density in the usual fashion. Clearly, a uniform density assigns infinite measure usual fashion. Clearly, a uniform density assigns infinite measure on the parameter space. More generally, if *g*(*Θ*) is any non- negative function defined on the parameter space Θ∈ Ω such that* g*(*Θ*)≠0 , then *g*(*Θ*) is called a “prior quasi-density.” Here, the integral ∫*g*(*Θ*) *dΘ* may or may not converge. Such densities have been used in recent years by Bayesian theoreticians and Wallace (1959) calls a prior quasi-density *g*(*Θ*) “admissible” with respect to a density *f( x : Θ )* defined for *x ∈ X* if:

almost everywhere on X. For every prior quasi-density * g*(*Θ*) which is admissible with respect to *f( x : Θ )*, there is defined a density on *g*(*Θ)* as follows;

If *g*(θ) or some constant multiple of *g*(θ) is a proper prior density, then by Bayes’s theorem, *g** in (31) is clearly a posterior density for *θ* give *x*. If, however, ∫*g*(*Θ*) *dΘ =∞, *then *g*(θ) is merely a formal substitution in the Bayes’s theorem on probability stated in (31). Strictly speaking, the theorem does not apply anymore but is still a g* proper probability density on Ω , and thus, an enthusiastic Bayesian may proceed undeterred with his usual analysis by taking g* as a posterior density. To distinguish, however, Wallace (1959) calls such as g* a “week posterior density.” A life tester facing problems of applied nature need not bother himself about this terminology but certainly one has a right to seek some justification of the use of densities which assign infinite measures on the parameter spaces for one’s own aesthetic satisfaction. To provide one, if on a subset ,

**then we can interpret**

as a conditional density given that θ ∈ Ω_{1} and clearly, this density defined on _{ }Ω_{1} is normalized. The special case g(θ)= 1 that is, uniform prior over the real line has earlier been interpreted in this manner by Lindley (1965). Various other justifications for the use of prior quasi-densities have also been given. If over some important subset Ω_{1}_{ }of parameter points where the likelihood is effective, the quasi-density resembles the actual prior, the fact that g(θ) is not integrable over Ω- _{ }Ω_{1} , may not be important. Or if g(θ) leads to posteriors which provide Bayesian inferences that closely resemble those of classical inference, then such a density may be used as a satisfactory approximation to the unknown prior distribution. Further theoretical justifications have been provided in recent literature by Bayesian statisticians and the reader interested in theory may consult Stone (1958) and Wallace (1959) for the details. The object here is neither to focus our attention on the controversies raised by the uniform prior of Jeffreys nor to make propaganda in favor of the use of prior quasi-densities but to merely explore the consequences of using such densities in the context of life testing problems. We hope some of the results may be useful for applied workers too. Thus, for situations where the life tester has no prior information about the parameter, we may use the uniform prior or more generally, we examine the quasi-density:

Under the assumption of this prior quasi-density, the posterior density of *Θ* after having observed the sample of first *r* ordered observations out of a sample of size *n* , turns out to be an inverted gamma density

Thus, a comparison with (7) immediately reveals that the results for this case can be obtained from earlier results by simply letting α→o and β→∞. Under these limiting conditions the results at (8) and (10) yield for the Bayes estimate and its variance respectively:

The substitution α=0 above yields the corresponding results for the diffuse prior that is uniform over the positive real line. Further, it is evident that only for α=2, (34) reduces to the well-known MVU estimate T_{r} /_{r}, Epstein and Sobel (1953). Thus, for the family of quasi-densities considered at (32), the Bayes solution is identical with classical result only when the parameter is assumed to have a prior quasi-density 1/*Θ*^{2} over the positive real line. As regards the estimates of the reliability function when the life has no prior information, we let α→o and β→∞ and in (22) and (23). Thus, a uniform prior over the positive real line leads to the following expressions:

When the complete sample is available, the reliability estimate obtained here reduces to:

That closely resembles the classical MVU estimate:

Given by Pugh (1963) if

For the numerical example considered by Pugh, the Bayes estimate of the reliability function and its variance work out to be 0.8387 and 0.0069 respectively, whereas Pugh obtains a value 0.89 for the reliability estimate but he has not evaluated any expression for the precision of his estimate. One may note that Pugh’s result as well as ours is valid for only *n*>1.

Strictly speaking, Pugh’s result holds for 0< t < nx̄

and for,* t ≥ nx̄*

That is, the classical method does not provide any information about the reliability function for large values of the time parameter. However, this is not the case with the Bayesian estimate, and the result holds for all *t > 0*.

The estimating function

is a monotonically decreasing function of t with R̂(0)=1 and R̂(t)→0 as t →∞ . This is a desirable property.

Similar results may be arrived at under suitable limiting conditions from the expressions corresponding to the exponential and the inverted gamma prior densities. Thus, letting μ → 0 and *v→ 1* in (21), (22), (23) and (24) we obtain estimates corresponding to the prior quasi-density 1/Θ^{2} . On the other hand, for the exponential prior (3) if we let , λ → ∞ the results correspond to those for a diffuse prior that is uniform over (o,∞ ). Under the limiting conditions stated above the limiting behavior of the prior densities has been thoroughly discussed by Raiffa and Schlaifer (1961) and we do not reproduce them here. The estimate of the reliability function based on attribute life tests may be obtained under the assumption of a uniform prior over the positive real line by letting α→o and β→∞ Thus,

The variance of this estimate is obtained by subtracting from

The square of the expression at (38).

That comes out as:

** **

**Bayes Estimate of Both Parameters in a Two Parameter Exponential Distribtuion **

Exponential distribution is a widely used lifetime distribution which has appeared in the literature since the early 1800s. This distribution is one of the commonly used statistical distributions in practice. Epstein and Sobel (1953, 1954, 1955), Epstein (1954), Bartholomew (1957), Mendenhall (1958), Johnson, Kotz and Balakrishnan (1994, 1995), Lawless (2003) and others have discussed this distribution with applications.

As given in Sinha (1986), the pdf of two parameter exponential distribution is given by:

If y_{1},y_{2}………y_{n} be individual observations from an exponential distribution, then the likelihood function is given by

Where y_{1} is the first order statistic in the sample such that *Y**=(Y*_{1,}Y2,…….Y_{n})

Jeffrey (1961) and others make extensive use of improper prior pdf’s. A more general class of priors has been considered here as,

**Posterior density for μ**** and θ ****:**

According to Bayes theorem, we have

Since, Posterior density ∝ (Prior density *likelihood)

That is

From the equations (42) and (43), the posterior density of μ and θ is given by

Where K is normalizing constant and is given by

Þ

**Marginal Posterior densities for μ**** and **** θ**

The marginal posterior density of μ is given by

The marginal posterior density of θ is given by

**BAYES ESTIMATES FOR μ**** AND θ**

The posterior estimate of μ is given by

The posterior estimates of θ is given by

**Numerical and Graphical Results**

Nineteen military personnel carriers failed in services for one reason or the other at the following mileages: 162, 200, 271, 302, 393, 508, 539, 629, 706, 777, 884, 1008, 1101, 1182, 1463, 1603, 1984, 2355 and 2880 miles. Posterior estimates μ of and θ are given in Table-I. The graphical representation for marginal posterior densities of μ and are θ shown in figures 1 and 2 respectively.

**Table 1: ****Posterior estimates of μ**** and θ**** under different priors**

**Figure 1**

**Figure 2**

The posteriors of μ and θ are plotted in figures 1 & 2 respectively. The posteriors μ are quite robust for varying in the prior P

(μ, θ) ∞ (1/θ^{c}) while the posteriors of θ are less robust.

**References**

- Bartholomew, D.J. (1957): “A problem in life testing”, JASA, Vol. 52, 350-355.
- Bhattacharya, S.K. and Holla, M. S. (1965): “On a life test distribution with stochastic deviations in the mean”. Ann. Stat. Math. 17, 97-104.
- Bhattacharya, S.K. (1962) : “On a probit analogue used in a life test based on Weibull distribution,” Aust. Jour. Stat. 4, 101-5.
- Bhattacharya, S.K. (1967) : “Bayesian approach to life testing and reliability estimation”, JASA, vol. 62, 48 -62.
- Donato, M C and E, Mellilli (2000) : “Some new results for Dirichelt priors”, The Annals of Statistics, Vol. 28, No. 5.
- Ehrenfeld, S.(1962) : “Some experimental design problems in life testing,” JASA, 57, 668-79.
- Epstein, B. and Sobel, M. (1953) : “Life testing,” JASA, 48, 486-502.
- Erdelyi, A. et al(1953): “Higher transcendental functions,” Vol. II, McGraw-Hill Book Company, Inc. N. Y.
- Govindarajulu, Z.(1964) : “A supplement to Mendenhalls bibliography on life testing and related topics,” JASA, 59, 1231-91.
- Jeffrey, H. (1961): “Theory of Probability”. Oxford University Press.
- Johnson, N.L., Kotz, S., and Balakrishnan, N. (1994) : “Continuous Univariate Distributions”, Vol. 1. John Wiley & Sons, New York.
- Johnson, N.L., Kotz, S., and Balakrishnan, N. (1995):” Continuous Univariate Distributions”, Vol. 2. John Wiley & Sons, New York.
- Karlin, S. and Rubin, H.(1956):”The theory of decision procedures for distributions with monotone likelihood ratio,” Ann. Math. Stat. 27, 272-99.
- Lawless, J.F. (2003) :” Statistical Models and Methods for Lifetime Data”, John Willey and Sons Inc., NY.
- Lindley, D. V.(1965) : “Introduction to probability and Statistics from a Bayesian view- point,” Part II, Inference, Cambridge, University Press.
- Mathew Stephens (2000) : “Bayesian analysis of mixture models with an unknown number of components – an alternatives to reversible jump methods”, the Annals of Statistics, Vol. 28, No.1, 40-74.
- Mendenhall, W. (1958): “A bibiliography on life testing and related topics”, Biometrika 45, 521-543.
- Pugh, E. L.(1963): “The best estimate of reliability in the exponential case” Opeartions Research (U.S.A.), 11, 57-61.
- Raiffa, H. and Schlaifer, R. (1961) : “Applied statistical decision theory”, Harvard University Press.
- Sinha, S.K.(1986) : “Reliability and life testing,” Willey Eastern Limited New Delhi.
- Stone, M. (1965) : “Right haar measure for convergence in probability to quasi-posterior distributions,” Ann. Math. Stat., 36, 440-53.
- Wallace, D. L.(1959): “Conditional confidence level properties,” Ann. Math. Stats. 30, 864-76.
- Wetherill, G.B.(1961) : “Bayesian sequential analysis”, Biometrika 48, 281-92.

(Visited 262 times, 1 visits today)

## 489 comments

Pingback: Google

Pingback: Google

Pingback: seo agency

Pingback: Venture Point Network

Pingback: most powerful vibrator

Pingback: jack rabbit vibrator

Pingback: bangal ka jadu

Pingback: so real dildo

Pingback: auto detailing

Pingback: Bond cleaning services

Pingback: Robin trimmer carburetor

Pingback: label tudung

Pingback: g spot toy

Pingback: max results pump

Pingback: Bond back clean Melbourne

Pingback: exterminators near me for mice

Pingback: vibrating butt plug

Pingback: gay sex toys

Pingback: best gay sex toys

Pingback: inflatable butt pug

Pingback: butt plug

Pingback: Butterfly Vibrator

Pingback: 9 inch dildo

Pingback: Candles

Pingback: adam and eve

Pingback: best end of lease cleaning Melbourne

Pingback: Top Sports Stories and Sports Headlines

Pingback: kala jadu

Pingback: best selling vibrator

Pingback: jelly dong

Pingback: g spot vibrator

Pingback: Magic Wand Massager

Pingback: Rabbit Vibrator

Pingback: Hand Held Massager

Pingback: Waterproof Vibrator

Pingback: Under Bed Restraints

Pingback: good head kit for him

Pingback: sex toys lubricant

Pingback: penis extension

Pingback: Cheap bond back cleaning Melbourne

Pingback: Sripatum

Pingback: app pc download for windows

Pingback: adam & eve delay spray

Pingback: pc games for windows 8

Pingback: sex restraint toys

Pingback: robert

Pingback: PA divorce forms

Pingback: Boom lift repair florida

Pingback: gay sex toys

Pingback: sex outfit

Pingback: couples

Pingback: free download for pc windows

Pingback: http://mer99furniture.com

Pingback: pc games for windows xp

Pingback: rabbit sex toys

Pingback: physical intimacy

Pingback: sex toys

Pingback: free download for windows 8

Pingback: http://americanairconditioninginc.com/

Pingback: free app for pc download

Pingback: pink vibrator

Pingback: suction cup vibrator

Pingback: adam and eve sex toys

Pingback: Foam dipped martial arts karate sparring gear

Pingback: education

Pingback: clitoral stimulators

Pingback: bondage toy

Pingback: male sex slave

Pingback: How to be a Better Lover

Pingback: flex vibrator’s bendabl

Pingback: hair loss clinic Singapore

Pingback: makeup vanity with mirror

Pingback: chiptuning files

Pingback: vibrator review

Pingback: pussy masturbator

Pingback: All Girth

Pingback: clitoris vibrator

Pingback: adam and eve sex toys

Pingback: couples toys

Pingback: خرید دوربین مدار بسته

Pingback: Fuel injector

Pingback: penis ring

Pingback: sex toys

Pingback: g-spot massager

Pingback: anal sex vibrators

Pingback: magic rabbit attachment

Pingback: best clit vibrator

Pingback: sex toys review

Pingback: jbeatz

Pingback: head honcho sex toy

Pingback: Adam and Eve Haul

Pingback: adam and eve adult toys

Pingback: Penis Pump Review

Pingback: Cock Ring

Pingback: bangladesh men's stone washing sewing suit manufacturer

Pingback: Lovers Toys

Pingback: climax

Pingback: Adam and Eve Big Rack Stroker

Pingback: rubber ducky vibrators

Pingback: chemise with garter

Pingback: skin feel dildo

Pingback: anal trainer kit

Pingback: app download for windows

Pingback: games for pc download

Pingback: fermented cabbage juice

Pingback: Inland Empire Ca

Pingback: anal toy review

Pingback: hand held massager

Pingback: Adamandeve.com

Pingback: Amy Scott

Pingback: glass sex toys

Pingback: dildo

Pingback: adam and eve

Pingback: What is the Purpose of Ben Wa Balls

Pingback: Pocket Stroker

Pingback: penis pump review

Pingback: Stocks to buy

Pingback: Black Belt

Pingback: egg vibrator

Pingback: anal sex toys

Pingback: Head Honcho Review

Pingback: robert

Pingback: tratamento de Alcoolismo

Pingback: Tunderkert

Pingback: nirvana 350 triple stimulator

Pingback: ترمیم مو

Pingback: sex role play

Pingback: realistic sex toys

Pingback: best female arousal products

Pingback: samuraiTV

Pingback: vpn خريد انلاين

Pingback: Century original wavemaster

Pingback: trade alerts

Pingback: adam and eve sex toys

Pingback: oral sex

Pingback: gay toys

Pingback: خرید اپل ایدی

Pingback: php ecommerce

Pingback: sex toys

Pingback: adam and eve

Pingback: sex toys

Pingback: Relationship

Pingback: best wand massager

Pingback: polythene manufacturer

Pingback: cartons

Pingback: handrail protection

Pingback: تردمیل خانگی

Pingback: magnetic label tape

Pingback: خرید اپل ایدی معتبر امریکا

Pingback: kala jadu

Pingback: adam and eve sex toys shop

Pingback: اجاره روزانه ویلا اطراف کرج

Pingback: خرید اپل آیدی رایگان

Pingback: A\u0026E Easy ‘O’ Red Rocket

Pingback: پویاوب

Pingback: remote control vibrator

Pingback: realistic vibrating dildo

Pingback: sex toys for couples

Pingback: sex toys

Pingback: bullet vibe

Pingback: best cream for sex

Pingback: vibrating tongue ring amazon

Pingback: ترمیم مو بیماران

Pingback: prostate massage

Pingback: urgent care los angeles

Pingback: http://pouyagym.com

Pingback: Light weight black martial arts gi

Pingback: full version pc games download

Pingback: Burn My Name

Pingback: revize jerabu

Pingback: 2pllogistics

Pingback: Tam Coc Vietnam

Pingback: ノロウイルス

Pingback: coca cola cup

Pingback: Download Free DJ Finesse Mixtapes

Pingback: Download Free DJ TY Boogie Mixtapes

Pingback: Download Free TY Boogie Mixtapes

Pingback: indianapolis

Pingback: adobe products

Pingback: auction marketplace

Pingback: etoro

Pingback: how cholesterol levels predict your health

Pingback: php download script

Pingback: XXXL sizes in black martial arts uniforms

Pingback: g-spot

Pingback: gay sex

Pingback: راهنمای خرید دوربین مداربسته

Pingback: best rabbit vibrator

Pingback: accounting

Pingback: Dildo

Pingback: Butt plugs

Pingback: sex whip

Pingback: triple action vibrator

Pingback: couples massage

Pingback: دوربین مدار بسته

Pingback: Anal toys

Pingback: Nipple Toys

Pingback: Cock ring

Pingback: bangal ka jadu

Pingback: Mikemaid7447

Pingback: bangal ka jadu

Pingback: driving tips

Pingback: adidas martial arts uniform

Pingback: masturbator sex toy

Pingback: Vincent Simmons

Pingback: fermented pickles

Pingback: magic wand massager

Pingback: sex position card game

Pingback: all american whopper

Pingback: strap-on dildo

Pingback: how to prepare for anal sex

Pingback: مشاوره حقوقی

Pingback: how to use anal lube

Pingback: http://www.bartarkafsab.com/

Pingback: خريد اپل ايدي

Pingback: door protection

Pingback: Cheap neodymium magnets

Pingback: double sided dildo

Pingback: نصب دوربین های مدار بسته

Pingback: sex therapy kit

Pingback: adidas Flex WTF dobock

Pingback: طراحی سایت شرکتی

Pingback: toys for couples

Pingback: dildo

Pingback: https://klout.com/#/~970525733406142492

Pingback: ترمیم مو و کاشت مو

Pingback: نصب و اجرای دوربین مداربسته

Pingback: فروش دوربین های مدار بسته مخفی

Pingback: Lil Wayne

Pingback: singer

Pingback: طراحی سایت تالار پذیرایی

Pingback: bdsm kit

Pingback: نصب دوربین های مدار بسته

Pingback: فروش دوربین های مدار بسته مخفی

Pingback: ترمیم مو و کاشت مو

Pingback: ترمیم مو، کاشت مو ، مو طبیعی

Pingback: portable sex swing

Pingback: adam and eve

Pingback: adammale.com

Pingback: تعمیر یخچال

Pingback: Évora

Pingback: TV/Film

Pingback: دوربین مدار بسته پویابین

Pingback: بهترین مارک دوربین مدار بسته

Pingback: Survival Ebooks

Pingback: آموزش نصب دوربین های مدار بسته

Pingback: ebooks with resale rights

Pingback: sex toys

Pingback: camera with date and time

Pingback: hot confessions

Pingback: Garden Bridges

Pingback: آموزش نصب دوربین های مدار بسته

Pingback: نصب دوربین های مدار بسته

Pingback: دوربین مداربسته

Pingback: prostate cancer

Pingback: cock ring review

Pingback: anal sex toy

Pingback: best jobs for working at home

Pingback: http://www.magnetic.co.uk/Category/diy-magnetic-secondary-glazing-kit

Pingback: heat sealer bags

Pingback: Ban Gioc Tours

Pingback: iron stair parts

Pingback: دوربین مدار بسته پویابین

Pingback: نصب و اجرای دوربین مداربسته

Pingback: Anti Slip Tape Primer and Coating

Pingback: ام دی اف

Pingback: http://www.shrc.asia/index.php?option=com_k2&view=itemlist&task=user&id=1125556

Pingback: pc games for windows 7

Pingback: #channelling

Pingback: free download for windows pc

Pingback: Chocolate dipped strawberries

Pingback: coffee beans Hawaii best gourmet kona

Pingback: app download for pc

Pingback: pc games for windows 8

Pingback: نصب و اجرای دوربین مداربسته

Pingback: apps for pc download

Pingback: pc app free download

Pingback: فروش دوربین های مدار بسته مخفی

Pingback: دوربین مدار بسته پویابین

Pingback: full apps download

Pingback: بهترین مارک دوربین مدار بسته

Pingback: pc games for windows 10

Pingback: pc games free download for pc windows

Pingback: آموزش نصب دوربین های مدار بسته

Pingback: free download for windows pc

Pingback: play free pc games

Pingback: LED Glass Wall

Pingback: Wholesale Gucci Bags

Pingback: http://mucangchaitours.com

Pingback: نصب دوربین های مدار بسته

Pingback: anti slip tape

Pingback: نصب و اجرای دوربین مداربسته

Pingback: فروش دوربین های مدار بسته مخفی

Pingback: دوربین مدار بسته پویابین

Pingback: بهترین مارک دوربین مدار بسته

Pingback: sapatenis osklen mercado livre

Pingback: 동화세상에듀코

Pingback: Florida Resident Database

Pingback: fall fashion sale

Pingback: fall fashion sale

Pingback: fall fashion sale

Pingback: home facial treatment machine

Pingback: בלוג גיימינג

Pingback: Band

Pingback: romance

Pingback: send email without limit

Pingback: دوربین مدار بسته

Pingback: دوربین مدار بسته ارزان

Pingback: دوربین های مدار بسته

Pingback: work from home jobs 2018

Pingback: دوربین مدار بسته ارزان

Pingback: دوربین های مدار بسته

Pingback: LELO

Pingback: remote control vibrator

Pingback: candy dildo

Pingback: Slim Dildo

Pingback: boots

Pingback: shoes

Pingback: 脱毛

Pingback: 脱毛

Pingback: خرید ، فروش دوربین مدار بسته

Pingback: Penn Valley taxi to Sacramento airport

Pingback: sex toys for gay men

Pingback: doubledildo

Pingback: how to use a double ended dildo

Pingback: دوربین مدار بسته بی سیم وایرلس

Pingback: Best Kona Coffee Online

Pingback: best kona coffee online

Pingback: VOD

Pingback: دوربین دنده عقب

Pingback: remote butt plug

Pingback: misting toy cleaner

Pingback: دوربین مدار بسته ارزان

Pingback: adamandeve.com

Pingback: orgasm

Pingback: دوربین های مدار بسته

Pingback: خرید ، فروش دوربین مدار بسته

Pingback: دوربین مدار بسته آنالوگ

Pingback: activation key

Pingback: دوربین مدار بسته بی سیم وایرلس

Pingback: دوربین دنده عقب

Pingback: Adam and Eve Coupon

Pingback: دوربین مدار بسته آنالوگ

Pingback: دوربین مدار بسته هایک ویژن

Pingback: فروش دوربین مداربسته

Pingback: استخدام شرکت دوربین مداربسته

Pingback: دوربین مدار بسته هایک ویژن

Pingback: دوربین مدار بسته هایک ویژن

Pingback: dual penetration

Pingback: how does a pocket pussy work

Pingback: penis sleeve

Pingback: Pocket Pussies

Pingback: How to Use a Penis Pump

Pingback: Canada Sex Toys

Pingback: Using a Rabbit Vibrator

Pingback: what is the best sex toy

Pingback: adam and eve sex toys shop

Pingback: g spot vibrator

Pingback: suction cup dildo

Pingback: تردمیل باشگاهی

Pingback: m88th

Pingback: دوچرخه ثابت

Pingback: Lion's Head tour

Pingback: اسکی فضایی الپتیکال

Pingback: Pet Products

Pingback: sex toys kit

Pingback: تردمیل خانگی

Pingback: adam and eve shopping

Pingback: morocco sahara desert tours

Pingback: تجهیزات ورزشی پویاجیم

Pingback: adam and eve free shipping code

Pingback: adam eve deal

Pingback: پک دوربین مداربسته ahd

Pingback: foam dipped awma sparring head

Pingback: ghchs hac

Pingback: تولید محتوا

Pingback: بازاریابی شبکه های اجتماعی

Pingback: دوربین مدار بسته دیجی کالا

Pingback: Brazilian Hair

Pingback: rsform pro download

Pingback: ارزانترین دوربین مدار بسته

Pingback: porn movie

Pingback: porn movie

Pingback: Hair Sisters Wholesale

Pingback: Milky Way Hair Wholesale

Pingback: Remy Hair Extensions Wholesale

Pingback: Hair Extensions Wholesale

Pingback: Peruvian Hair Wholesale

Pingback: Milky Way Hair Wholesale

Pingback: Malaysian Hair Wholesale

Pingback: Indian Hair Wholesale

Pingback: شرکت دوربین مدار بسته

Pingback: Virgin Hair Wholesale

Pingback: psicologo adicciones

Pingback: فروش دوربین مدار بسته

Pingback: angeles & angels

Pingback: ผ้าญี่ปุ่นพาหุรัด

Pingback: hackers for hire

Pingback: دوربین مدار بسته وای فای

Pingback: بهترین دوربین دنده عقب

Pingback: prediksi bola

Pingback: Footwear for police officers

Pingback: قیمت دوربین آنالوگ

Pingback: Anal Beads Sex

Pingback: vibrating bullet

Pingback: bullet sex toy

Pingback: vaginal tightening

Pingback: suterh

Pingback: Craft foam sheets

Pingback: هایک ویژن

Pingback: بازاریابی محتوا

Pingback: بازاریابی شبکه اجتماعی

Pingback: kona coffee

Pingback: read about

Pingback: ترمیم مو و کاشت مو

Pingback: Cloud Archiving

Pingback: دوربین مدار بسته مخفی

Pingback: دوربین مدار بسته خودرو

Pingback: martial arts and gymnastics mixed

Pingback: بازاریابی محتوا

Pingback: vibrator review

Pingback: kona coffee

Pingback: خدمات تولید محتوا

Pingback: how to use a rabbit vibrator

Pingback: armytrix exhaust tuning price

Pingback: بازاریابی شبکه اجتماعی

Pingback: realistic vibrating dildo

Pingback: kegel exercises

Pingback: oral sex tips for him

Pingback: male stroker

Pingback: realistic vibrating

Pingback: mini vibrator sex

Pingback: magic mini massager

Pingback: name

Pingback: adidas yeezy 350 v2

Pingback: دوربین مدار بسته پویابین

Pingback: what is a rabbit vibrator

Pingback: silicone finger vibrator

Pingback: how much is a penis pump

Pingback: Buy Vigra

Pingback: anal sex toys

Pingback: خرید دوربین مداربسته

Pingback: wholesale web hosting

Pingback: quinny

Pingback: bedroom lingerie

Pingback: کابینت mdf

Pingback: icicles butt plug

Pingback: butt vibrator

Pingback: کفسابی

Pingback: dildos suction cup

Pingback: dildos strap on

Pingback: Airport Limo Mineapolis

Pingback: پک آماده دوربین مداربسته

Pingback: power stud rod

Pingback: best bunny vibrator

Pingback: best cryptocurrency

Pingback: kona coffee

Pingback: دزدگیر اماکن

Pingback: Airport limo Minneapolis

Pingback: como mantener una buena ereccion

Pingback: هوشمند سازی ساختمان

Pingback: تعرفه تولید محتوا

Pingback: طراحی سایت

Pingback: mini vibrator

Pingback: vibrating ring

Pingback: digital marketing

Pingback: فروش دوربین مدار بسته

Pingback: خرید اپل ایدی

Pingback: cheap hotels west palm beach

Pingback: real life sex dolls

Pingback: خرید اپل آیدی

Pingback: wild orchid evolved

Pingback: online mobile number for sms receive

Pingback: Virtual Visa and MasterCard.credit card

Pingback: female sex toys

Pingback: oplata smieciowa warszawa

Pingback: adam and eve sex toys

Pingback: adult toy websites

Pingback: adam and eve discount coupons

Pingback: adam eve welcome kit

Pingback: adam et eve online

Pingback: adam n eve lingerie

Pingback: sex toys for single women

Pingback: best selling vibrator

Pingback: تولید محتوا

Pingback: best rated sex toys

Pingback: anal sex lube

Pingback: ترمیم مو

Pingback: where to get a dildo