Independent Discrete Random Variables Research Articles

A distribution on ℤ via perturbing the Laplace distribution with applications to finance and health data

This paper is devoted to a new two‐parameter distribution on having an original additive stochastic structure. It is defined by the sum of two independent discrete random variables, where one is defined based on the Laplace distribution and the other on a transformed version of a distribution that can be viewed as a relative binomial distribution. It can thus also be viewed as a perturbed version of the Laplace distribution. We present some interesting properties of the proposed distribution, such as its moments (ordinary, non‐central, incomplete moments, etc.), characteristic function, skewness, and kurtosis, among others. The distribution is also applied to practical datasets from finance and health studies, where it is seen to perform better than some of the existing models defined on .

Estimation of P(X ≤ Y) for discrete distributions with non-identical support

Abstract The Uniformly Minimum Variance Unbiased (UMVU) and the Maximum Likelihood (ML) estimations of R = P(X ≤ Y) and the associated variance are considered for independent discrete random variables X and Y. Assuming a discrete uniform distribution for X and the distribution of Y as a member of the discrete one parameter exponential family of distributions, theoretical expressions of such quantities are derived. Similar expressions are obtained when X and Y interchange their roles and both variables are from the discrete uniform distribution. A simulation study is carried out to compare the estimators numerically. A real application based on demand-supply system data is provided.

On the maximum entropy of a sum of independent discrete random variables

Пусть $X_1,…, X_n $ - независимые случайные величины, принимающие значения в алфавите $\{0,1,…,r\} $, и $S_n=\sum_{i=1}^nX_i$. Теорема Шеппа-Олкина утверждает, что в бинарном случае ($r=1$) энтропия Шеннона случайной величины $S_n$ максимальна, когда все $X_i$ равномерно распределены, т.е. являются бернуллиевскими с параметром $1/2$. Стремясь обобщить эту теорему на случай конечных алфавитов, мы получаем нижнюю границу для максимума энтропии случайной величины $S_n$ и доказываем, что она точна в некоторых частных случаях. В дополнение к этим частным случаям приводится еще один довод в поддержку гипотезы о том, что полученная граница представляет собой оптимальное значение для всех $n$, $r$, т.е. что $H(S_n)$ максимальна, когда $X_1,…,X_{n-1}$ равномерно распределены на $\{0,r\}$, в то время как функция распределения масс случайной величины $X_n$ является смесью (с явным образом определенными ненулевыми весами) равномерных распределений на $\{0,r\}$ и $\{1,…,r-1\}$.

On the Maximum Entropy of a Sum of Independent Discrete Random Variables

Let $ X_1, \ldots, X_n $ be independent random variables taking values in the alphabet $ \{0, 1, \ldots, r\} $, and $ S_n = \sum_{i = 1}^n X_i $. The Shepp--Olkin theorem states that, in the binary case ($ r = 1 $), the Shannon entropy of $ S_n $ is maximized when all the $ X_i $'s are uniformly distributed, i.e., Bernoulli(1/2). In an attempt to generalize this theorem to arbitrary finite alphabets, we obtain a lower bound on the maximum entropy of $ S_n $ and prove that it is tight in several special cases. In addition to these special cases, an argument is presented supporting the conjecture that the bound represents the optimal value for all $ n, r $, i.e., that $ H(S_n) $ is maximized when $ X_1, \ldots, X_{n-1} $ are uniformly distributed over $ \{0, r\} $, while the probability mass function of $ X_n $ is a mixture (with explicitly defined non-zero weights) of the uniform distributions over $ \{0, r\} $ and $ \{1, \ldots, r-1\} $.

Renewed Looks at the Distribution of a Sum of Independent or Dependent Discrete Random Variables and Related Problems

Butler and Stephens (2017) have investigated the exact and approximate distributions of a sum S of independent binomial random variables with different probabilities. They used a convolution approach to find the exact distribution, whereas they heavily used the moments and cumulants to find approximations. We propose two different approaches. First, we show that the moment generating function (MGF) approach is easier and simpler to implement for finding the exact distribution of S. We also provide approximations for the distribution of S using large-scale computer simulations based on one million independent replications each. Such exact and approximate distributions are very close to analogous values reported in Butler and Stephens (2017). We show versatility of our approaches by including both exact and approximate distributions for the sum S of independent multinomial, geometric and other discrete random variables.

On the joint entropy of $d$-wise-independent variables

How low can the joint entropy of $n$ $d$-wise independent (for $d\geq 2$) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than $p$, for $p< 1$)? This question has been posed and partially answered in a recent work of Babai [{Entropy versus pairwise independence} (preliminary version), {http://people.cs.uchicago.edu/~laci/papers/13augEntropy.pdf}, 2013]. In this paper we improve some of his bounds, prove new bounds in a wider range of parameters and show matching upper bounds in some special cases. In particular, we prove tight lower bounds for the min-entropy (as well as the entropy) of pairwise and three-wise independent balanced binary variables for infinitely many values of $n$.

A Novel Method for Increasing the Entropy of a Sequence of Independent, Discrete Random Variables

In this paper, we propose a novel method for increasing the entropy of a sequence of independent, discrete random variables with arbitrary distributions. The method uses an auxiliary table and a novel theorem that concerns the entropy of a sequence in which the elements are a bitwise exclusive-or sum of independent discrete random variables.

Maximum Likelihood for Matrices with Rank Constraints

Maximum likelihood estimation is a fundamental optimization problem in statistics. Westudy this problem on manifolds of matrices with bounded rank. These represent mixtures of distributionsof two independent discrete random variables. We determine the maximum likelihood degree for a rangeof determinantal varieties, and we apply numerical algebraic geometry to compute all critical points oftheir likelihood functions. This led to the discovery of maximum likelihood duality between matrices ofcomplementary ranks, a result proved subsequently by Draisma and Rodriguez.

Stirling's Approximation for Central Extended Binomial Coefficients

We derive asymptotic formulas for central extended binomial coefficients, which are generalizations of binomial coefficients, using the distribution of the sum of independent discrete uniform random variables with the Central Limit Theorem and a local limit variant.

A pointwise approximation for random sums of independent discrete random variables

Let {Xn,i, i = 1, ..., n;n ∈ N} be a sequence of independent integer-valued random variables with success probabilities P (Xn,i = 1) = pn,i and P (Xn,i = 0) = 1 − pn,i − qn,i for every i ∈ {1, ..., n} and n ∈ N, where pn,i, qn,i ∈ (0, 1) and pn,i + qn,i ∈ (0, 1). Suppose that Nn for n ∈ N are positive integer-valued random variables and independent of the Xn,i’s. Let SNn = ∑Nn i=1Xn,i be random sums of a sequence of independent integer-valued random variables, and let λNn = ∑Nn i=1 pn,i. In this case, Hung and Giang [1] gave a pointwise bound for approximating the probability function of the random sums SNn by the Poisson probability function with mean λNn as follows:

Reliability and non-reliability studies of Poisson variables in series and parallel systems

In this paper, we have derived the distribution of the minimum and maximum of two independent Poisson random variables. A useful procedure for computing the probabilities is given and a total of four numerical examples are presented. Of these four examples, the first two are on the generated data and the other two are on the Champion League Soccer data in order to illustrate the model which is considered here. The hazard rate and the reversed hazard rate, of the minimum and maximum of two independent discrete random variables, are also obtained and their monotonicity is investigated. The results for the Poisson-distributed variables are obtained as special cases.

The probabilistic gradual covering location problem on a network with discrete random demand weights

We study the gradual covering location problem on a network with uncertain demand. A single facility is to be located on the network. Two coverage radii are defined for each node. The demand originating from a node is considered fully covered if the shortest distance from the node to the facility does not exceed the smaller radius, and not covered at all if the shortest distance is beyond the larger radius. For a distance between these two radii, the coverage level is specified by a coverage decay function. It is assumed that demand weights are independent discrete random variables. The objective of the problem is to find a location for the facility so as to maximize the probability that the total covered demand weight is greater than or equal to a pre-selected threshold value. We show that the problem is NP-hard and that an optimal solution exists in a finite set of dominant points. We develop an exact algorithm and a normal approximation solution procedure. Computational experiment is performed to evaluate their performance.

Monotonicity, Thinning, and Discrete Versions of the Entropy Power Inequality

We consider the entropy of sums of independent discrete random variables, in analogy with Shannon's Entropy Power Inequality, where equality holds for normals. In our case, infinite divisibility suggests that equality should hold for Poisson variables. We show that some natural analogues of the Entropy Power Inequality do not in fact hold, but propose an alternative formulation which does always hold. The key to many proofs of Shannon's Entropy Power Inequality is the behaviour of entropy on scaling of continuous random variables. We believe that R\'{e}nyi's operation of thinning discrete random variables plays a similar role to scaling, and give a sharp bound on how the entropy of ultra log-concave random variables behaves on thinning. In the spirit of the monotonicity results established by Artstein, Ball, Barthe and Naor, we prove a stronger version of concavity of entropy, which implies a strengthened form of our discrete Entropy Power Inequality.

The distribution of a linear combination of R independent discrete random variables

In this paper we consider a formula to get the exact number of nonnegative integer solution of the equality a1x1 + a2x2 + … + arxr = n where a1, a2, … , ar and n are fixed integers. Using the obtained formula, we provide a program to list the solutions for every n and a1, … , ar by Pascal compiler. We then obtain the distribution of an arbitrary linear combination of discrete random variables based on the proposed algorithm. We also apply the algorithm to obtain the Maximum Likelihood estimation of the parameters of the distribution. The accuracy of the algorithm was illustrated using various examples.

On the singularity probability of discrete random matrices

Let n be a large integer and M n be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this paper is to prove a general upper bound for the probability that M n is singular. For a constant 0 < p < 1 and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of M n satisfy this property, then the probability that M n is singular is at most ( p 1 / r + o ( 1 ) ) n . All of the results in this paper hold for any characteristic zero integral domain replacing the complex numbers. In the special case where the entries of M n are “fair coin flips” (taking the values + 1 , − 1 each with probability 1/2), our general bound implies that the probability that M n is singular is at most ( 1 2 + o ( 1 ) ) n , improving on the previous best upper bound of ( 3 4 + o ( 1 ) ) n , proved by Tao and Vu [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628]. In the special case where the entries of M n are “lazy coin flips” (taking values + 1 , − 1 each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that M n is singular is at most ( 1 2 + o ( 1 ) ) n , which is asymptotically sharp. Our method is a refinement of those from [Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223–240; Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628]. In particular, we make a critical use of the structure theorem from [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628], which was obtained using tools from additive combinatorics.

The probabilistic 1-maximal covering problem on a network with discrete demand weights

We discuss the probabilistic 1-maximal covering problem on a network with uncertain demand. A single facility is to be located on the network. The demand originating from a node is considered covered if the shortest distance from the node to the facility does not exceed a given service distance. It is assumed that demand weights are independent discrete random variables. The objective of the problem is to find a location for the facility so as to maximize the probability that the total covered demand is greater than or equal to a pre-selected threshold value. We show that the problem is NP-hard and that an optimal solution exists in a finite set of dominant points. We develop an exact algorithm and a normal approximation solution procedure. Computational experiment is performed to evaluate their performance.

Probability Generating Functions for Discrete Real-Valued Random Variables

The probability generating function is a powerful technique for studying the law of finite sums of independent discrete random variables taking integer positive values. For real-valued discrete random variables, the well-known elementary theory of Dirichlet series and the symbolic computation packages available nowadays, such as Mathematica 5, allow us to extend this technique to general discrete random variables. Being so, the purpose of this work is twofold. First, we show that discrete random variables taking real values, nonnecessarily integer or rational, may be studied with probability generating functions. Second, we intend to draw attention to some practical ways of performing the necessary calculations.

The β-Reliable Median on a Network with Discrete Probabilistic Demand Weights

This study extends the classical network median problem by considering the stochastic nature of demand. Assuming that the demand weights associated with nodes are independent discrete random variables, we introduce a chance-constrained programming model to define a β-reliable median of the network. It is shown that the β-reliable median problem is NP-hard. Exact solution procedures and a normal approximation algorithm are developed to search for the β-reliable median. Their performance is evaluated by computational experiments.

Cycle-transitive comparison of independent random variables

The discrete dice model, previously introduced by the present authors, essentially amounts to the pairwise comparison of a collection of independent discrete random variables that are uniformly distributed on finite integer multisets. This pairwise comparison results in a probabilistic relation that exhibits a particular type of transitivity, called dice-transitivity. In this paper, the discrete dice model is generalized with the purpose of pairwisely comparing independent discrete or continuous random variables with arbitrary probability distributions. It is shown that the probabilistic relation generated by a collection of arbitrary independent random variables is still dice-transitive. Interestingly, this probabilistic relation can be seen as a graded alternative to the concept of stochastic dominance. Furthermore, when the marginal distributions of the random variables belong to the same parametric family of distributions, the probabilistic relation exhibits interesting types of isostochastic transitivity, such as multiplicative transitivity. Finally, the probabilistic relation generated by a collection of independent normal random variables is proven to be moderately stochastic transitive.

Algorithms for computing the distributions of sums of discrete random variables

We present algorithms for computing the probablity density function of the sum of two independent discrete random variables, along with an implementation of the algorithm in a computer algebra system. Some examples illustrate the utility of this algorithm.