# Geometric distribution examples and solutions pdf

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- On estimating the parameter of a truncated geometric distribution

Documentation Help Center. A scalar input is expanded to a constant array with the same dimensions as the other input. The parameters in p must lie on the interval [0,1]. Suppose you toss a fair coin repeatedly, and a "success" occurs when the coin lands with heads facing up. What is the probability of observing exactly three tails "failures" before tossing a heads?

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B Yenagoa, Bayelsa, Nigeria. A two-parameter Rayleigh-geometric distribution with increasing-decreasing-increasing and strictly increasing hazard rate characteristics is reviewed. Various properties are discussed and expressed analytically. The estimation of the distribution parameters is studied by the method of maximum likelihood and validated by a simulation study.

Numerical examples based on two real data-sets on the waiting time in queue and CO 2 emissions are given. The Rayleigh-geometric distribution in this paper has a simpler analytical expression compared to the pre-existing distributions with different parameterizations.

In recent times, there is increasing attempt by several researchers from different academic spheres to define new probability distributions for appropriate modeling of various complex real-life phenomena. Some of the new probability distributions are quite flexible in that they result in some other well-defined probability distributions when their parameter s are set to certain values.

For example, Mahmoudi and Shiran introduced a four-parameter distribution known as the exponentiated Weibull-geometric EWG distribution.

The EWG distribution is a compound mixture of the exponentiated Weibull EW distribution due to Mudholkar and Srivastava and the geometric distribution. This paper is devoted to studying a probability distribution that had appeared in previous studies in slightly different forms. The distribution in its present form is simpler in expression and it has among other appealing characteristics an increasing hazard rate.

The increasing hazard rate is a popular and useful concept in life testing and so many other areas of applied probability and statistics, for example; many real-life phenomena particularly those that are related to electronic components, devices, and machines exhibit the increasing hazard rate property due to regular use and wear.

Moreover, the distribution under study is identified to be suitable for modeling queueing and CO 2 emissions data. The remaining part of this paper is in the following order. Section 2 introduces the distribution, defines some related functions like the reliability and hazard rate functions, and discusses some limit behaviors, Section 3 gives some basic properties of the distribution, Section 4 presents the estimation of parameters, Section 5 illustrates the usefulness and flexibility of the distribution by two real data-sets, and Section 6 gives the concluding remarks.

The RGD is a special case of the geometric generalized family of distributions and the physical interpretation of the exponential-geometric distribution EGD due to Adamidis and Loukas can be extended to the RGD. To the best of our knowledge, there is no paper focusing on either the direct derivation of the RGD or dedicated to the discussion of its properties and inferential issues as in this paper but, as we mentioned earlier, the RGD has apparently emerged in the literature as a special case of some well-known distributions.

For instance:. Apart from generating lookup tables for fractiles, the quantile function has also been used to study several mathematical properties of a probability distribution. An alternative to the classical measure of skewness and kurtosis due to Galton and Moors , respectively is based on the quantile function and they are known as the Galton's skewness and Moors' kurtosis.

It is clear from Fig. Mixture representations of Equations 1 and 2 are required to obtain the properties of the RGD, the representations are based on the generalized binomial series and this idea is used throughout the remaining sections.

Rewriting Equations 1 and 2 in terms of Equation 3 we obtain. The mean E X of the RGD could be obtained from Equation 6 by setting k to 1 and other higher order moments can equally be obtained by the appropriate substitution of k. Like the mean and median, the mode is used to describe the distribution of a random variable.

The mode of a continuous random variable X say, is the value of X at which its PDF has a local maximum value. The local maximum value of the RGD is obtained as the solution of. Equation 11 does not have any analytical solution in terms of x and must be calculated numerically. The result in Table 2 shows that the mode of the RGD is always within the interval of 0 , 1 and decreases with the increasing values of p.

The application of Lorenz and Bonferroni curves are popularly known in Economics as indices for measuring inequality in terms of income and poverty distribution in a given population and they have also gained widespread utility in reliability engineering, actuarial science, and medicine. We omit the detail for the derivations of Equation 12 , 13 , and 14 because they are similar to that of Equation 6.

The total observed Fisher information matrix can be approximated by. For a given set of observations, the matrix in Equation 12 can be obtained after the convergence of the Newton-Raphson procedure via the nlm function in R software. In this section, we study the performance of the method of maximum likelihood estimation in estimating the parameters of the RGD using Monte-Carlo simulation for different sample sizes n and different parameter values.

It is easy to verify from these results that the MLEs approximates the true parameter values as n increases. In this section, we use a real data-set to demonstrate the performance of the Rayleigh-geometric distribution in modeling real-life data-set. The total CO 2 emissions from the consumption and flaring of fossil fuels in millions of metric tons of carbon dioxide were compiled by the US Department of energy www.

The data is available in the R package called asbio R Core Team, The second data is on the waiting time minutes before service of bank customers that were reported in Merovci and Elbatal The summary statistics for the two data-sets are listed in Table 4.

We compare the fitting performance of the RGD in modeling the CO 2 and the waiting time data with the following competing one-parameter, two-parameter, and three-parameter distributions:. Lindley distribution due to Lindley EGD due to Adamidis and Loukas The analytical expressions for the goodness-of-fit measures are:.

Akaike information criterion AIC due to Akaike Bayes information criterion BIC due to Schwarz However, in practice, the answer to the question of which between the two distributions RGD and WGD should be chosen is readily handy by the popular rule of parsimony which in principle favors the RGD distribution. The hypothesis is based on 0. Parameter estimates, standard errors in bracket , and the log-likelihood of the fitted models for the CO 2 data.

Parameter estimates, standard errors in bracket , and the log-likelihood of the fitted models for the waiting time data. Plots of the estimated reliability function for the RGD red lines and WGD blue lines superimposed on the empirical reliability function for the waiting time data a and CO 2 data b. Anwar and Bibi applied the HLGW distribution to the waiting time data and the fit of the HLGW distribution was compared with those of six competing one, two, and three-parameter distributions including the Weibull distribution.

Based on the results from the distributions fittings in Table 4 of Anwar and Bibi the HLGW distribution indicate a better performance than the other distributions in modeling the waiting time data. Here, we want to point out that the two-parameter RGD in this paper offers a better fit to the waiting time data than the three-parameter HLGW distribution because the log-likelihood of the RGD In terms of parsimony, the two-parameter RGD, gamma distribution, and EE distribution are generally better options for modeling the waiting time data than the HLGW distribution.

The statistical literature is lacking the complete representation of the exact parametrization of the Rayleigh-geometric distribution RGD as in this paper. On this note, we present a detailed account of the statistical properties of the two-parameter RGD with two applications to illustrate its possible utility. The RGD was obtained by mixing the geometric distribution and the Rayleigh distribution; moreover, the Rayleigh distribution is identified as the limiting case of the RGD when the mixing parameter p approaches zero.

The hazard rate function of the distribution contains some important shapes of the hazard rate of some lifetime phenomena that are commonly encountered in practice, they are; increasing-decreasing-increasing and strictly increasing shapes. Estimation of the distribution parameters was carried out by the method of maximum likelihood and a numerical study show that the maximum likelihood estimation method provides good estimates for the distribution parameters.

Two real-life examples based on the modeling of waiting time in bank queue and CO 2 emissions in Afghanistan were given to demonstrate the usefulness and fitting prowess of the RGD. We hope that this paper would encourage the application of the RGD in areas such as hydrology, reliability analysis, and meteorology. Idika E. Okorie: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Anthony C. Akpanta, Johnson Ohakwe, David C. Chikezie, Chris U. Onyemachi: Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

National Center for Biotechnology Information , U. Journal List Heliyon v. Published online Aug 5. Akpanta , b Johnson Ohakwe , c David C. Chikezie , b and Chris U. Onyemachi b. David C. Chris U. Author information Article notes Copyright and License information Disclaimer.

Okorie: ku. Published by Elsevier Ltd. Abstract A two-parameter Rayleigh-geometric distribution with increasing-decreasing-increasing and strictly increasing hazard rate characteristics is reviewed.

Keywords: Mathematics, Hazard rate, Rayleigh distribution, Geometric distribution, Queueing analysis, Maximum likelihood. Introduction In recent times, there is increasing attempt by several researchers from different academic spheres to define new probability distributions for appropriate modeling of various complex real-life phenomena. Open in a separate window. Figure 1. Relationship with other distributions To the best of our knowledge, there is no paper focusing on either the direct derivation of the RGD or dedicated to the discussion of its properties and inferential issues as in this paper but, as we mentioned earlier, the RGD has apparently emerged in the literature as a special case of some well-known distributions.

For instance: 1. Theory 3. Quantile function and random number generation Apart from generating lookup tables for fractiles, the quantile function has also been used to study several mathematical properties of a probability distribution. Theorem 3. Corollary 3. Figure 2. Useful expansions Mixture representations of Equations 1 and 2 are required to obtain the properties of the RGD, the representations are based on the generalized binomial series and this idea is used throughout the remaining sections.

Table 1 Some numerical values of the mean and variance of the RGD. Figure 3. Mode Like the mean and median, the mode is used to describe the distribution of a random variable. Table 2 Some numerical values of the mode of the RGD. Lorenz and Bonferroni curves The application of Lorenz and Bonferroni curves are popularly known in Economics as indices for measuring inequality in terms of income and poverty distribution in a given population and they have also gained widespread utility in reliability engineering, actuarial science, and medicine.

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For a fixed number of trials n, the binomial distribution always behaves in the same way: as a function of k, it is monotone increasing up to a certain point m after which perhaps with an exception of the next point. Worksheets PDF - The bigger portal for free educational material. The probability of at least two successes in four trials of a binomial experiment in which. If p is the probability of success and q is the probability of failure in a binomial trial, then the expected number of successes in n trials i. Use this binomial probability calculator to easily calculate binomial cumulative distribution function and probability mass given the probability on a single trial, the number of trials and events.

(b) A failure with probability q = 1 − p. 3. Repeated trials are independent. X = number of trials to first success. X is a GEOMETRIC RANDOM VARIABLE. PDF.

## On estimating the parameter of a truncated geometric distribution

What is the probability that it takes five games until you lose? You throw darts at a board until you hit the center area. You want to find the probability that it takes eight throws until you hit the center. What values does X take on? She decides to look at the accident reports selected randomly and replaced in the pile after reading until she finds one that shows an accident caused by failure of employees to follow instructions.

B Yenagoa, Bayelsa, Nigeria. A two-parameter Rayleigh-geometric distribution with increasing-decreasing-increasing and strictly increasing hazard rate characteristics is reviewed. Various properties are discussed and expressed analytically. The estimation of the distribution parameters is studied by the method of maximum likelihood and validated by a simulation study. Numerical examples based on two real data-sets on the waiting time in queue and CO 2 emissions are given.

*I have written several articles about how to work with continuous probability distributions in SAS.*

#### 1. Introduction

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