Independent Exponential Random Variables Research Articles

Stochastic Comparisons of Largest-Order Statistics and Ranges from Marshall–Olkin Bivariate Exponential and Independent Exponential Variables

Sample range and the associated functions such as survival function and mean residual life function have found many important applications in the reliability field. In this work, we establish some results that are in two different directions. In the first part, we establish some conditions for comparing the largest-order statistics (in the sense of mean residual life order) arising from bivariate Marshall–Olkin exponential distribution. Then, in the second part, we present some sufficient conditions for comparing sample ranges (in the sense of usual stochastic order and reversed hazard rate order) arising from independent exponential random variables.

A new bivariate distribution with uniform marginals

Starting from three independent exponential random variables we have generated a bivariate random vector (U, V) having the marginal distributions as standard uniform. The joint distribution function and the survival function have been derived along with the moment generating function. We have also given an expression for the joint moment of order (r, s). The distributions of different functions of U and V have been derived. Different dependence measures between U and V have also been calculated. The reliability of the underlying stress-strength model has been obtained as an application of the distribution. A simulation exercise has been carried out to check for the goodness-of-fit of the model. We have also calculated the relative errors in different reliability measures under the assumption that the variables U and V are independent when actually they are not.

Remarks on the Papers by Coelho-Barros et al. (2017), Usman et al. (2021) and Obeid and Kadry (2022)

We would like to point out that the formula for the cumulative distribution given in Coelho-Barros et al. (2017) and a similar version of it given in Usman et al. (2021) are not cumulative distribution functions as these functions do not satisfy the one or more necessary and sufficient conditions for a function to be a cumulative distribution function. We would also like to mention that formulas for the cumulative distribution functions of product and ratio of two independent Pareto and Exponential random variables given by Obeid and Kadry (2022) are not cumulative distribution functions either. We do not believe that these formulas can be fixed to be cumulative distribution functions. In this short article, we provide mathematical justification in support of these claims.

Dependence comparisons of order statistics in the proportional hazards model

Let $X_1, \ldots, X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda _1, \ldots, \lambda _n$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lambda = \sum _{i=1}^{n} \lambda _i/n$. Also let $X_{1:n} \lt \cdots \lt X_{n:n}$ and $Y_{1:n} \lt \cdots \lt Y_{n:n}$ be their associated order statistics. It is proved that for $1\le i \lt j \le n$, the generalized spacing $X_{j:n} - X_{i:n}$ is more dispersed than $Y_{j:n} - Y_{i:n}$ according to dispersive ordering and for $2\le i \le n$, the dependence of $X_{i:n}$ on $X_{1:n}$ is less than that of $Y_{i:n}$ on $Y_{1 :n}$, in the sense of the more stochastically increasing ordering. This dependence result is also extended to the proportional hazard rates (PHR) model. This extends the earlier work of Genest et al. [(2009)]. On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592] who proved this result for $i =n$.

Distributions of the Product and Ratio of Two Independent Pareto and Exponential Random Variables

Distributions of the Product and Ratio of Two Independent Pareto and Exponential Random Variables

DETERMINATION OF VARIANCE OF TIME TO RECRUITMENT WITH INTER-DECISION TIME AS GEOMETRIC PROCESS WHEN THE THRESHOLD HAS THREE COMPONENTS

In this paper, the problem of time to recruitment is analyzed for a single grade manpower system using an univariate CUM policy of recruitment. Assuming policy decisions and exits occur at different epochs, wastage of manpower due to exits form a sequence of independent and identically distributed exponential random variables, the inter-decision times form a geometric process and inter-exist time form an independent and identically distributed random variable. The breakdown threshold for the cumulative wastage of manpower in the system has three components which are independent exponential random variables. Employing a different probabilistic analysis, analytical results in closed form for system characteristics are derived

STOCHASTIC MODEL FOR TIME TO RECRUITMENT IN A SINGLE GRADE MANPOWER SYSTEM WITH TWO TYPES OF INTERDECISION TIMES WHEN THE BREAKDOWN THRESHOLD HAS THREE COMPONENTS

In this paper, the problem of time to recruitment is analyzed for a single grade manpower system in which attrition takes place due to two types of policy decisions where this classification is done according to intensity of attrition, it form an ordinary renewal process. Assuming (i) policy decisions and exits occur at different epochs (ii) wastage of manpower due to exits and wastage due to frequent breaks taken by the personnel working in the manpower system separately form a sequence of independent and identically distributed exponential random variables with different means and (iii) breakdown threshold for the cumulative wastage of manpower in the system has three components which are independent exponential random variables. A stochastic model is constructed and the variance of the time to recruitment is obtained using an univariate CUM policy of recruitment. Employing a different probabilistic analysis, analytical results in closed form for system characteristics are derived.

On the density for sums of independent exponential, Erlang and gamma variates

This paper re-examines the density for sums of independent exponential, Erlang and gamma random variables. By using a divided difference perspective, the paper provides a unified approach to finding closed-form formulae for such convolutions. In particular, the divided difference perspective for sums of Erlang variates suggests a new approach to finding the density for sums of independent gamma variates using fractional calculus.

Exponential and Hypoexponential Distributions: Some Characterizations

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n≥2, X1,X2,…,Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villaseñor (2013) for a particular convolution of two random variables.

Stochastic Modeling as a Method of Arriving at Higher Frequencies: An Application to κ Estimation

AbstractWe present a framework for stochastically modeling the Fourier spectrum of the noisy seismic recording based on the fundamental assumption that the latter constitutes a random rather than a deterministic quantity. First we demonstrate (mathematically and through simulations) that, under the stochastic‐signal assumption, the periodogram ordinates of the noisy recording can be considered independent exponential random variables with a frequency‐dependent mean. Using this finding, the estimation of seismological parameters is translated into a well‐defined maximum likelihood (ML) problem, allowing a fast, accurate, and robust solution. Although the proposed ML methodology constitutes a general estimation framework that we believe can improve any spectral analysis, here we specifically apply it to the high‐frequency attenuation parameter (κ), crucial for understanding rock ground motion. Other seismological parameters could be estimated by appropriately adapting the theoretical model and frequency band used. The greatest advantage of the proposed method is its ability to account for the presence of noise rather than simply try to avoid it, as is the case with all conventional κ approaches and many other spectral analysis approaches. This means that the proposed technique achieves acceptable results even for very low signal‐to‐noise ratios (SNR), thus pushing the boundary of what can be considered usable recording quality. The technique's superior performance in estimation is demonstrated through a series of experiments involving both synthetic and real seismic recordings. Our results also indicate that, compared to this new “noise‐modeling” approach, conventional “noise‐avoiding” approaches are likely to systematically underestimate the “true” kappa value in recordings of moderate‐to‐low SNR.

Sequential fixed-accuracy confidence intervals for the stress–strength reliability parameter for the exponential distribution: two-stage sampling procedure

In this paper, a two-stage sequential estimation procedure is considered to construct fixed-accuracy confidence intervals of the reliability parameter R under the stress–strength model when the stress and strength are independent exponential random variables with different scale parameters. The exact distribution of the total sample size, explicit formulas for the expected value, and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Performances of the proposed methodology are investigated with the help of simulations. Finally, using three pairs of real datasets, the procedure is clearly illustrated. We used MATLAB in order to implement computations, simulation studies, and real data analysis.

Permutation Tests for Infection Graphs

We formulate and analyze a novel hypothesis testing problem for inferring the edge structure of an infection graph. In our model, a disease spreads over a network via contagion or random infection, where the times between successive contagion events are independent exponential random variables with unknown rate parameters. A subset of nodes is also censored uniformly at random. Given the observed infection statuses of nodes in the network, the goal is to determine the underlying graph. We present a procedure based on permutation testing, and we derive sufficient conditions for the validity of our test in terms of automorphism groups of the graphs corresponding to the null and alternative hypotheses. Our test is easy to compute and does not involve estimating unknown parameters governing the process. We also derive risk bounds for our permutation test in a variety of settings, and relate our test statistic to approximate likelihood ratio testing and maximin tests. For graphs not satisfying the necessary symmetries, we provide an additional method for testing the significance of the graph structure, albeit at a higher computational cost. We conclude with an application to real data from an HIV infection network. Supplementary materials for this article are available online.

A stochastic inventory system with two modes of service under N-policy

We consider a two server stochastic inventory system under N-policy. Demand for the items occurs one at a time according to a Poisson process. Server1 is always available and Server2 is activated on the accumulation of N(>1) demands in the system. As soon as the services of all waiting demands are completed, server2 becomes idle and reactivate only on the accumulation of N units. Assume that service times of both servers are independent and non-identical exponential random variables. We derived the joint probability distribution of the waiting customers, server state and inventory level in the steady state and computed various system performance measures. A cost analysis is carried out and numerical illustrations are provided.

ON A TWO-SERVER RELIABILITY SYSTEM WITH ONE SERVER IDLE BELOW A THRESHOLD

In this work, we study a two-server k-out-of-n: G repairable system with identical components. Lifetimes of components follow an exponential distribution with parameter λ. For the repair of failed components, we consider two servers called server1 and server2 and the repair times are independent exponential random variables with parameters μ1 and μ2 (μ1 1), whereas both servers are active above that threshold. That is, activating server2 only when the number of failed components reaches the threshold N and when it falls below that threshold, switch off the server. It will again reactivate only on the accumulation of N units. We obtained the system state distributions in finite time and the steady-state, by analysing a continuous time Markov chain and derived several other characteristics of the system such as system reliability, mean time to failure, mean busy period of the servers, etc. Also, we investigated a control problem and proved that the total expected cost of the system is convex in N and hence global minimum exists. Some numerical illustrations also provided.

Inference for the linear combination of two independent exponential random variables based on fuzzy data

In this article, we have derived the distribution of a linear combination of two independent exponential random variables. The parameter estimates of the proposed distribution are obtained by using the maximum likelihood estimation method and the method of moments from fuzzy data. The findings in this paper show that estimation expertise is still valuable to any organization based on the precise and certain information. The proposed research consisted in developing an estimation technique using fuzzy logic and this is often measured in terms of linguistic values, which is very helpful and beneficial  for the certain and precise results.

On Stress-Strength Interval-System Reliability with Applications in Heart Conditions

The random variable X represents the stress placed on the system by the operating environment and random variable Y represents the strength of the system. A system is able to perform its intended function if its strength is greater than the stress imposed upon it. Reliability of the system is defined as the probability that the system is strong enough to overcome the stress. That is, R = P(Y >X). In other words, reliability is the probability that the strengths of the unit are greater than the stresses. The stress-strength model has found interests in many applications include mechanical engineering and human heart monitoring conditions. The interval-system is defined as a system with a series of chance events that occur in a given interval of time. A k-out-of-n interval-system is a system with a series of n events in a given interval of time which successes (or functions) if and only if at least k of the events succeed (function). In short, the k-out-of-n interval-system is an interval-system which successes if and only if at least k of n events succeeds. The stress-strength reliability inference of the interval-system with a series of n independent events that occurs in a given interval of time is considered. The reliability of the interval-system is the probability that at least k out of n events in a given interval of time succeed. This paper derives uniform minimum variance unbiased and maximum likelihood reliability estimates of k-out-of-n interval-system based on stress-strength inference events where X (stress) and Y (strength) are independent two-parameter exponential random variables. An application in human heart conditions to illustrate the results is discussed.

From Coalescing Random Walks on a Torus to Kingman’s Coalescent

Let $\mathbb{T}^d_N$, $d\ge 2$, be the discrete $d$-dimensional torus with $N^d$ points. Place a particle at each site of $\mathbb{T}^d_N$ and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two particles meet, they coalesce into one. Denote by $C_N$ the first time the set of particles is reduced to a singleton. Cox [6] proved the existence of a time-scale $\theta_N$ for which $C_N/\theta_N$ converges to the sum of independent exponential random variables. Denote by $Z^N_t$ the total number of particles at time $t$. We prove that the sequence of Markov chains $(Z^N_{t\theta_N})_{t\ge 0}$ converges to the total number of partitions in Kingman's coalescent.

On a queueing-inventory system with advanced reservation and cancellation for the next K time frames ahead: the case of overbooking

We analyse the evolution of a system designed for reservation of some items in advance (for example, seats in aircrafts or trains or bus) by customers arriving at random moments. The reservation has to be done by the server in one of the K time frames. At the beginning of the pth time frame, the inventoried items in it (as well as those sold from it earlier) have life time distribution which is a p-fold convolution of a phase-type distribution with itself, for $$1 \le p \le K.$$ Cancellation of reserved items is possible before the expiry of their life. Distributions characterizing the service and inter-cancellation times are assumed to be independent exponential random variables and the customer arrivals are according to a Markovian arrival process. The number of items for reservation, available at the beginning of each time frame, is finite. If, at the commencement of service of a customer, the item in the required time frame is not available, the reservation may still be possible, through overbooking. Overbooking up to a maximum fixed level is permitted for each time frame. If, for the required day, the overbooked item is available, the customer is served the same. If this too is not available, he is asked to give alternatives. If none of his alternatives can be met, he is provided with a reservation for the time frame (day) for which one is available. If that too is not available, then he will have to wait until the expiry of one time frame; in the last case all remaining customers will have to wait. On expiry of one phase distribution, the time frames are renumbered and a new time frame with a K-fold convolution of the phase-type distribution is added $$(0 \leftarrow 1 \leftarrow 2 \leftarrow \cdots \leftarrow K-1 \leftarrow K \leftarrow K+1).$$ All overbooked customers present in the recently expired time frame are provided with a reservation in the newly added time frame (which has, at that epoch, a life time of a K-fold convolution of the phase-type distribution). This system is analysed and illustrated through numerical experiments. The special case of Poisson arrival, coupled with blocking of arrivals when all time frames $$1, 2, \ldots , K$$ are overbooked, is shown to yield a product form solution. For this case, an appropriate cost function is constructed and its properties investigated numerically.

Parameter estimates of general failure rate model: A Bayesian approach

The failure rate function plays an important role in studying the lifetime distributions in reliability theory and life testing models. A study of the general failure rate model r ( t ) = a + b t θ − 1 , under squared error loss function, taking a and b independent exponential random variables, has been performed in the literature. In this article, we consider a and b not necessarily independent. The estimates of the parameters a and b under squared error loss, linex loss and entropy loss functions are obtained, and further examined using simulated and real data sets. • A general failure rate model r ( t ) = a + b t θ − 1 has been studied when a and b are not necessarily independent. • Bayes’ estimators of the parameters a and b, under symmetric and asymmetric loss functions, have been obtained. • The monotonicity of the estimators has been studied through simulated as well as real data sets

Exponential order statistics and some combinatorial identities

It is known that the k-th () order statistic from the unit exponential distribution can be represented as a sum of independent exponential random variables. We present a proof of this result based on Laplace transform. Also, computing the Laplace transform of the k-th order statistic in two different ways and equating them, we derive several interesting combinatorial identities. A probabilistic interpretation of these identities and their generalizations are also given.