A NEW ONE PARAMETER RAYLEIGH MAXWELL DISTRIBUTION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this article, a new one-parameter lifetime distribution has been suggested. The distribution is a two-component finite mixture of Rayleigh and Maxwell distribution. The various statistical properties of the distribution such as moments, skewness, kurtosis, moment generating function, and characteristics function have been discussed. Survival function and hazard function are also studied. The maximum likelihood estimate for the unknown parameter under the proposed model is derived. Finally, the model is fitted to a real-life failure data, and outcomes were compared with some standard statistical probability distributions.


INTRODUCTION
The temperature of a physical system is determined by the velocity of the particles (atoms or molecules) contributing to the system. These particles have different velocities and they also constantly changes due to collisions among themselves. Maxwell (1860) [13] showed that the energy of such particles exhibits a certain probability pattern. Boltzmann (1872) [4] later simplified the probability distribution and investigated its physical origin.
Though the initial development of Maxwell -Boltzmann distribution revolves around gas particles, Tyagi and Bhattacharya (1989a, b) [17] [18] for the first time considered this distribution to model lifetime data. They also explored its inferential properties under the Bayesian approach and reliability function. Chaturvedi and Rani (1998) [5] generalized the distribution through transformation on gamma random variate. Poddar and Roy (2003) [15]  The Rayleigh distribution was introduced by Lord Rayleigh (1880) [16] to study a problem in the field of acoustics. The distribution is related to several well-known distributions such as Chi-Square, Exponential, Gamma, and Weibull. The distribution has a wide range of application and hence extensive work has been done in various fields of science and technology (Johnson et. al. 1994) [11].
A physical system, for example, industrial equipment or vehicles are comprised of many individual and vital parts. All of these parts of a system may exhibit a completely different failure pattern. Considering a single probability distribution to explore the survival function of such a system may not always give us the desired outcome. A finite mixture of some known and suitable probability distributions can help in understanding the sub-populations of a system with different properties. Finite mixtures are found to be useful in various fields of physics, chemistry, biology, and social sciences.
In this study, a finite mixture of Rayleigh and Maxwell (RMM) distribution is proposed.
The various statistical properties of the mixture are discussed. The parameter of the proposed mixture is estimated under the maximum likelihood method. Finally, the mixture is fitted to a real-life data set.

Let us consider a two component mixture of Rayleigh distribution with parameter a and
Maxwell-Boltzmann distribution with parameter a with their mixing components 1 1+a and a 1+a respectively. The probability distribution of the new distribution can be written as The corresponding cdf is given by is the lower incomplete gamma integral.

Moments of RMM Distribution.
The r th raw moment is given by Replacing particular values of r (r = 1,2,3,4) in (3) we get the first four raw moments as The corresponding central moments are

Harmonic Mean. The harmonic mean of RMM distribution is
2.1.3. Mode of RMM distribution. The mode of a distribution is obtained by solving the equa- On simplification, the equation to obtain mode of RMM distribution reduces to

Survival Function. The survival function of our proposed RMM model is given by
Where,

Hazard Function. The hazard function of RMM distribution is obtained as
The reverse hazard rate function of RMM distribution is ; x > 0 (20)

INFERENTIAL PROCEDURES
The likelihood equation corresponding to (1) is The log likelihood function is Differentiating (22) w.r.t. a and equating to zero we get Solving (23) for a numerically, we can get the maximum likelihood estimate of the parameter.

APPLICATION
The proposed model is fitted to a data set related to the number of miles to the first major motor failure of 191 buses operated by a large city bus company (Davis [6]    Our proposed model is performing better in explaining the data set than the remaining distributions since the values of AIC, AICC, HQIC, and CAIC are less compare to Rayleigh, Maxwell -Boltzman, Gamma, Chi-square and Exponential distribution.

CONCLUSION
The superior performance of our proposed model can be confirmed from the different discrimination criteria since the best model is the one that gives the minimum values of those criteria. The distribution can be used in cases where we observed a high rate of failure as we move towards the mode of the data and then failure rate decreases drastically. The various statistical properties of the proposed model were also discussed. Further extension of the proposed model was also possible and will be studied in future work.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.