Theorizing Probability Distribution in Applied Statistics

This paper investigates into theoretical knowledge on probability distribution and the application of binomial, poisson and normal distribution. Binomial distribution is widely used discrete random variable when the trails are repeated under identical condition for fixed number of times and when there are only two possible outcomes whereas poisson distribution is for discrete random variable for which the probability of occurrence of an event is small and the total number of possible cases is very large and normal distribution is limiting form of binomial distribution and used when the number of cases is infinitely large and probabilities of success and failure is almost equal.

A note on negative λ-binomial distribution

In this paper, we introduce one discrete random variable, namely the negative λ-binomial random variable. We deduce the expectation of the negative λ-binomial random variable. We also get the variance and explicit expression for the moments of the negative λ-binomial random variable.

Information Loss Due to the Data Reduction of Sample Data from Discrete Distributions

In this paper, we study the information lost when a real-valued statistic is used to reduce or summarize sample data from a discrete random variable with a one-dimensional parameter. We compare the probability that a random sample gives a particular data set to the probability of the statistic’s value for this data set. We focus on sufficient statistics for the parameter of interest and develop a general formula independent of the parameter for the Shannon information lost when a data sample is reduced to such a summary statistic. We also develop a measure of entropy for this lost information that depends only on the real-valued statistic but neither the parameter nor the data. Our approach would also work for non-sufficient statistics, but the lost information and associated entropy would involve the parameter. The method is applied to three well-known discrete distributions to illustrate its implementation.

Peluang, Peubah Acak Diskrit, dan Sebaran Peluang Peubah Acak Diskrit

Probability to learn someone's chance in getting or winning an event. In the discrete random variable is more identical to repeated experiments, to form a pattern. Discrete random variables can be calculated as the probability distribution by calculating each value that might get a certain probability value.

Exact cash-account deflator for the G2++ model

In this paper, we shall propose a Monte Carlo simulation technique applied to a G2++ model: even when the number of simulated paths is small, our technique allows to find a precise simulated deflator. In particular, we shall study the transition law of the discrete random variable :[Formula: see text] in the time span [Formula: see text] conditional on the observation at time [Formula: see text], and we apply it in a recursive way to build the different paths of the simulation. We shall apply the proposed technique to the insurance industry, and in particular to the issue of pricing insurance contracts with embedded options and guarantees.

A note on characteristic function for Bernstein polynomials involving special numbers and polynomials

The aim of this present paper is to establish and study generating function associated with a characteristic function for the Bernstein polynomials. By this function, we derive many identities, relations and formulas relevant to moments of discrete random variable for the Bernstein polynomials (binomial distribution), Bernoulli numbers of negative order, Euler numbers of negative order and the Stirling numbers.

λ-Analogues of Stirling polynomials of the first kind and their applications

Recently, λ-analogues of Stirling numbers of the first kind were studied. In this paper, we introduce, as natural extensions of these numbers, λ-Stirling polynomials of the first kind and r-truncated λ-Stirling polynomials of the first kind. We give recurrence relations, explicit expressions, some identities, and connections with other special polynomials for those polynomials. Further, as applications, we show that both of them appear in an expression of the probability mass function of a suitable discrete random variable, constructed from λ-logarithmic and negative λ-binomial distributions.

Efficient Optimal Approximation of Discrete Random Variables for Estimation of Probabilities of Missing Deadlines

We present an efficient algorithm that, given a discrete random variable X and a number m, computes a random variable whose support is of size at most m and whose Kolmogorov distance from X is minimal. We present some variants of the algorithm, analyse their correctness and computational complexity, and present a detailed empirical evaluation that shows how they performs in practice. The main application that we examine, which is our motivation for this work, is estimation of the probability of missing deadlines in series-parallel schedules. Since exact computation of these probabilities is NP-hard, we propose to use the algorithms described in this paper to obtain an approximation.