Computation of Some Important Distributional Properties

Description

Generally, most of the packages specify the probability density function, cumulative distribution function, quantile function, and random numbers generation of the probability distributions. The present package allows to compute some important distributional properties, including the first four ordinary and central moments, Pearson's coefficient of skewness and kurtosis, the mean and variance, coefficient of variation, median, and quartile deviation at some parametric values of several well-known and extensively used probability distributions.

Details

Maintainers

Muhammad Imran <imranshakoor84@yahoo.com>

Author(s)

Christophe Chesneau <christophe.chesneau@unicaen.fr>, Muhammad Imran <imranshakoor84@yahoo.com>, M.H Tahir <mht@iub.edu.pk> and Farrukh Jamal <farrukh.jamal@iub.edu.pk>.

Compute the distributional properties of the beta distribution

Description

Compute the first four ordinary moments, central moments, mean, variance, Pearson's coefficient of skewness, kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the beta distribution.

Usage

d_beta(alpha, beta)

Arguments

Details

The following is the probability density function of the beta distribution:

f(x)=\frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}x^{\alpha-1}\left(1-x\right)^{\beta-1},

where 0\leq x\leq1, \alpha > 0 and \beta > 0.

Value

d_beta gives the first four ordinary moments, central moments, mean, variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the beta distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Gupta, A. K., & Nadarajah, S. (2004). Handbook of beta distribution and its applications. CRC Press.

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). Beta distributions. Continuous univariate distributions. 2nd ed. New York, NY: John Wiley and Sons, 221-235.

See Also

d_kum

Examples

d_beta(2,2)

Compute the distributional properties of the beta exponential distribution

Description

Compute the first four ordinary moments, central moments, mean, variance, Pearson's coefficient of skewness, kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the beta exponential distribution.

Usage

d_bexp(lambda, alpha, beta)

Arguments

Details

The following is the probability density function of the beta exponential distribution:

f(x)=\frac{\lambda e^{-\beta\lambda x}}{B(\alpha,\beta)}\left(1-e^{-\lambda x}\right)^{\alpha-1},

where x > 0, \alpha > 0, \beta > 0 and \lambda > 0.

Value

d_bexp gives the first four ordinary moments, central moments, mean, variance, Pearson's coefficient of skewness, kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the beta exponential distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Nadarajah, S., & Kotz, S. (2006). The beta exponential distribution. Reliability Engineering & System Safety, 91(6), 689-697.

See Also

d_beta

Examples

d_bexp(1,1,0.2)

Compute the distributional properties of the Birnbaum-Saunders distribution

Description

Compute the first four ordinary moments, central moments, mean, variance, Pearson's coefficient of skewness, kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Birnbaum-Saunders distribution.

Usage

d_bs(v)

Arguments

Details

The following is the probability density function of the Birnbaum-Saunders distribution:

f(x)=\frac{x^{0.5}+x^{-0.5}}{2vx}\phi\left(\frac{x^{0.5}-x^{-0.5}}{v}\right),

where x > 0 and v > 0.

Value

d_bs gives the first four ordinary moments, central moments, mean, variance, Pearson's coefficient of skewness, kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Birnbaum-Saunders distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Chan, S., Nadarajah, S., & Afuecheta, E. (2016). An R package for value at risk and expected shortfall. Communications in Statistics Simulation and Computation, 45(9), 3416-3434.

See Also

d_normal

Examples

d_bs(5)

Compute the distributional properties of the Burr XII distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Burr XII distribution.

Usage

d_burr(k, c)

Arguments

Details

The following is the probability density function of the Burr XII distribution:

f(x)=kcx^{c-1}\left(1+x^{c}\right)^{-k-1},

where x > 0, c > 0 and k > 0.

Value

d_burr gives the first four ordinary moments, central moments, mean, variance, Pearson's coefficient of skewness, kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Burr XII distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Rodriguez, R. N. (1977). A guide to the Burr type XII distributions. Biometrika, 64(1), 129-134.

Zimmer, W. J., Keats, J. B., & Wang, F. K. (1998). The Burr XII distribution in reliability analysis. Journal of Quality Technology, 30(4), 386-394.

See Also

d_kburr

Examples

d_burr(2,10)

Compute the distributional properties of the Chen distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Chen distribution.

Usage

d_chen(k, c)

Arguments

Details

The following is the probability density function of the Chen distribution:

f(x)=ckx^{k-1}e^{x^{k}}e^{c-ce^{x^{k}}},

where x > 0, c > 0 and k > 0.

Value

d_chen gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Chen distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Chen, Z. (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics & Probability Letters, 49(2), 155–161.

See Also

d_wei, d_EE, d_EW

Examples

d_chen(0.2,0.2)

Compute the distributional properties of the Chi-squared distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the (non-central) Chi-squared distribution.

Usage

d_chi(n)

Arguments

Details

The following is the probability density function of the (non-central) Chi-squared distribution:

f(x)=\frac{1}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}x^{\frac{n}{2}-1}e^{-\frac{x}{2}},

where x > 0 and n > 0.

Value

d_chi gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the (non-central) Chi-squared distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Ding, C. G. (1992). Algorithm AS275: computing the non-central chi-squared distribution function. Journal of the Royal Statistical Society. Series C (Applied Statistics), 41(2), 478-482.

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John Wiley & Sons.

See Also

d_gamma

Examples

d_chi(2)

Compute the distributional properties of the exponential distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the exponential distribution.

Usage

d_exp(alpha)

Arguments

Details

The following is the probability density function of the exponential distribution:

f(x)=\alpha e^{-\alpha x},

where x > 0 and \alpha > 0.

Value

d_exp gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the exponential distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Balakrishnan, K. (2019). Exponential distribution: theory, methods and applications. Routledge.

Singh, A. K. (1997). The exponential distribution-theory, methods and applications, Technometrics, 39(3), 341-341.

See Also

d_wei, d_EE

Examples

d_exp(2)

Compute the distributional properties of the exponential extension distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the exponential extension distribution.

Usage

d_nh(alpha, beta)

Arguments

Details

The following is the probability density function of the exponential extension distribution:

f(x)=\alpha\beta(1+\alpha x)^{\beta-1}e^{1-(1+\alpha x)^{\beta}},

where x > 0, \alpha > 0 and \beta > 0.

Value

d_nh gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the exponential extension distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Nadarajah, S., & Haghighi, F. (2011). An extension of the exponential distribution. Statistics, 45(6), 543-558.

See Also

d_exp

Examples

d_nh(0.5,1)

Compute the distributional properties of the exponentiated Weibull distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the exponentiated Weibull distribution.

Usage

d_EW(a, beta, zeta)

Arguments

Details

The following is the probability density function of the exponentiated Weibull distribution:

f(x)=a\zeta\beta^{-\zeta}x^{\zeta-1}e^{-\left(\frac{x}{\beta}\right)^{\zeta}}\left[1-e^{-\left(\frac{x}{\beta}\right)^{\zeta}}\right]^{a-1},

where x > 0, a > 0, \beta > 0 and \zeta > 0.

Value

d_EW gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the exponentiated Weibull distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. (2013). The exponentiated Weibull distribution: a survey. Statistical Papers, 54, 839-877.

See Also

d_EE, d_wei

Examples

d_EW(1,1,0.5)

Compute the distributional properties of the exponentiated exponential distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the exponentiated exponential distribution.

Usage

d_EE(alpha, beta)

Arguments

Details

The following is the probability density function of the exponentiated exponential distribution:

f(x)=\alpha\beta e^{-\alpha x}\left(1-e^{-\alpha x}\right)^{\beta-1},

where x > 0, \alpha > 0 and \beta > 0.

Value

d_EE gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the exponentiated exponential distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Nadarajah, S. (2011). The exponentiated exponential distribution: a survey. AStA Advances in Statistical Analysis, 95, 219-251.

Gupta, R. D., & Kundu, D. (2007). Generalized exponential distribution: Existing results and some recent developments. Journal of Statistical Planning and Inference, 137(11), 3537-3547.

See Also

d_EW, d_wei, d_exp

Examples

d_EE(5,2)

Compute the distributional properties of the F distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the F distribution.

Usage

d_F(alpha, beta)

Arguments

Details

The following is the probability density function of the F distribution:

f(x)=\frac{1}{B(\frac{\alpha}{2},\frac{\beta}{2})}\left(\frac{\alpha}{\beta}\right)^{\frac{\alpha}{2}}x^{\frac{\alpha}{2}-1}\left(1+\frac{\alpha}{\beta}x\right)^{-\left(\frac{\alpha+\beta}{2}\right)},

where x > 0, \alpha > 0 and \beta > 0.

Value

d_F gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the F distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John Wiley & Sons.

See Also

d_gamma

Examples

d_F(2,10)

Compute the distributional properties of the Frechet distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Frechet distribution.

Usage

d_fre(alpha, beta, zeta)

Arguments

Details

The following is the probability density function of the Frechet distribution:

f(x)=\frac{\alpha}{\zeta}\left(\frac{x-\beta}{\zeta}\right)^{-1-\alpha}e^{-(\frac{x-\beta}{\zeta})^{-\alpha},}

where x>\beta, \alpha>0, \zeta>0 and \beta\in\left(-\infty,+\infty\right). The Frechet distribution is also known as inverse Weibull distribution and special case of the generalized extreme value distribution.

Value

d_fre gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Frechet distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Abbas, K., & Tang, Y. (2015). Analysis of Frechet distribution using reference priors. Communications in Statistics-Theory and Methods, 44(14), 2945-2956.

See Also

d_wei

Examples

d_fre(5,1,0.5)

Compute the distributional properties of the gamma distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the gamma distribution.

Usage

d_gamma(alpha, beta)

Arguments

Details

The following is the probability density function of the gamma distribution:

f(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x},

where x > 0, \alpha > 0 and \beta > 0.

Value

d_gamma the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the gamma distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Burgin, T. A. (1975). The gamma distribution and inventory control. Journal of the Operational Research Society, 26(3), 507-525.

See Also

d_wei, d_naka

Examples

d_gamma(2,2)

Compute the distributional properties of the Gompertz distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Gompertz distribution.

Usage

d_gompertz(alpha, beta)

Arguments

Details

The following is the probability density function of the Gompertz distribution:

f(x)=\alpha e^{\beta x-\frac{\alpha}{\beta}\left(e^{\beta x}-1\right)},

where x > 0, \alpha > 0 and \beta > 0.

Value

d_gompertz gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Gompertz distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Soliman, A. A., Abd-Ellah, A. H., Abou-Elheggag, N. A., & Abd-Elmougod, G. A. (2012). Estimation of the parameters of life for Gompertz distribution using progressive first-failure censored data. Computational Statistics & Data Analysis, 56(8), 2471-2485.

See Also

d_fre

Examples

d_gompertz(2,2)

Compute the distributional properties of the Gumbel distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Gumbel distribution.

Usage

d_gumbel(alpha, beta)

Arguments

Details

The following is the probability density function of the Gumbel distribution:

f(x)=\frac{1}{\beta}e^{-(z+e^{-z})},

where z=\frac{x-\alpha}{\beta}, x\in\left(-\infty,+\infty\right), \alpha\in\left(-\infty,+\infty\right) and \beta > 0.

Value

d_gumbel gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Gumbel distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Gomez, Y. M., Bolfarine, H., & Gomez, H. W. (2019). Gumbel distribution with heavy tails and applications to environmental data. Mathematics and Computers in Simulation, 157, 115-129.

See Also

d_gompertz, d_fre

Examples

d_gumbel(1,2)

Compute the distributional properties of the inverse-gamma distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the inverse-gamma distribution.

Usage

d_ingam(alpha, beta)

Arguments

Details

The following is the probability density function of the inverse-gamma distribution:

f(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{-\alpha-1}e^{-\frac{\beta}{x}},

where x > 0, \alpha > 0 and \beta > 0.

Value

d_ingam gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the inverse-gamma distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Rivera, P. A., Calderín-Ojeda, E., Gallardo, D. I., & Gómez, H. W. (2021). A compound class of the inverse Gamma and power series distributions. Symmetry, 13(8), 1328.

Glen, A. G. (2017). On the inverse gamma as a survival distribution. Computational Probability Applications, 15-30.

See Also

d_gamma

Examples

d_ingam(5,2)

Compute the distributional properties of the Kumaraswamy Burr XII distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Kumaraswamy Burr XII distribution.

Usage

d_kburr(a, b, k, c)

Arguments

Details

The following is the probability density function of the Kumaraswamy Burr XII distribution:

f(x)=\frac{abkcx^{c-1}}{\left(1+x^{c}\right)^{k+1}}\left[1-\left(1+x^{c}\right)^{-k}\right]^{a-1}\left\{ 1-\left[1-\left(1+x^{c}\right)^{-k}\right]^{a}\right\} ^{b-1},

where x > 0, a > 0, b > 0, k > 0 and c > 0.

Value

d_kburr gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Kumaraswamy Burr XII distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Paranaiba, P. F., Ortega, E. M., Cordeiro, G. M., & Pascoa, M. A. D. (2013). The Kumaraswamy Burr XII distribution: theory and practice. Journal of Statistical Computation and Simulation, 83(11), 2117-2143.

See Also

d_kum, d_kexp

Examples

d_kburr(1.5,1,1,7)

Compute the distributional properties of the Kumaraswamy distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Kumaraswamy distribution.

Usage

d_kum(alpha, beta)

Arguments

Details

The following is the probability density function of the Kumaraswamy distribution:

f(x)=\alpha\beta x^{\alpha-1}\left(1-x^{a}\right)^{\beta-1},

where 0\leq x\leq1, \alpha > 0 and \beta > 0.

Value

d_kum gives the first four ordinary moments, central moments, mean, variance, Pearson's coefficient of skewness, kurtosis, coefficient of variation, median and quartile deviation at some parametric values based on the Kumaraswamy distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

El-Sherpieny, E. S. A., & Ahmed, M. A. (2014). On the kumaraswamy distribution. International Journal of Basic and Applied Sciences, 3(4), 372.

Mitnik, P. A. (2013). New properties of the Kumaraswamy distribution. Communications in Statistics-Theory and Methods, 42(5), 741-755.

Dey, S., Mazucheli, J., & Nadarajah, S. (2018). Kumaraswamy distribution: different methods of estimation. Computational and Applied Mathematics, 37, 2094-2111.

See Also

d_beta

Examples

d_kum(2,2)

Compute the distributional properties of the Kumaraswamy exponential distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Kumaraswamy exponential distribution.

Usage

d_kexp(lambda, a, b)

Arguments

Details

The following is the probability density function of the Kumaraswamy exponential distribution:

f(x)=ab\lambda e^{-\lambda x}\left(1-e^{-\lambda x}\right)^{a-1}\left\{ 1-\left(1-e^{-\lambda x}\right)^{a}\right\} ^{b-1},

where x > 0, a > 0, b > 0 and \lambda > 0.

Value

d_kexp gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Kumaraswamy exponential distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883-898.

See Also

d_kburr, d_kum

Examples

d_kexp(0.2,1,1)

Compute the distributional properties of the Kumaraswamy normal distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Kumaraswamy normal distribution.

Usage

d_kumnorm(mu, sigma, a, b)

Arguments

Details

The following is the probability density function of the Kumaraswamy normal distribution:

f(x)=\frac{ab}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right)\left[\Phi\left(\frac{x-\mu}{\sigma}\right)\right]^{a-1}\left[1-\Phi\left(\frac{x-\mu}{\sigma}\right)^{a}\right]^{b-1},

where x\in\left(-\infty,+\infty\right), \mu\in\left(-\infty,+\infty\right), \sigma > 0, a > 0 and b > 0. The functions \phi(.) and \Phi(.) , denote the probability density function and cumulative distribution function of the standard normal variable, respectively.

Value

d_kumnorm gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Kumaraswamy normal distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883-898.

See Also

d_kburr, d_kexp, d_kum

Examples

d_kumnorm(0.2,0.2,2,2)

Compute the distributional properties of the Laplace or double exponential distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Laplace distribution.

Usage

d_lap(alpha, beta)

Arguments

Details

The following is the probability density function of the Laplace distribution:

f(x)=\frac{1}{2\beta}e^{\frac{-|x-\alpha|}{\beta}},

where x\in\left(-\infty,+\infty\right), \alpha\in\left(-\infty,+\infty\right) and \beta > 0.

Value

d_lap gives the first four ordinary moments, central moments, mean, variance, Pearson's coefficient of skewness, kurtosis, coefficient of variation, median and quartile deviation at some parametric values based on the Laplace distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Cordeiro, G. M., & Lemonte, A. J. (2011). The beta Laplace distribution. Statistics & Probability Letters, 81(8), 973-982.

See Also

d_normal

Examples

d_lap(2,4)

Compute the distributional properties of the log-normal distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the log-normal distribution.

Usage

d_lnormal(mu, sigma)

Arguments

Details

The following is the probability density function of the log-normal distribution:

f(x)=\frac{1}{x\sigma\sqrt{2\pi}}e^{-\frac{\left(\log(x)-\mu\right)^{2}}{2\sigma^{2}}},

where x > 0, \mu\in\left(-\infty,+\infty\right) and \sigma > 0.

Value

d_lnormal gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the log-normal distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 1, Chapter 14. Wiley, New York.

See Also

d_normal

Examples

d_lnormal(1,0.5)

Compute the distributional properties of the logistic distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the logistic distribution.

Usage

d_logis(mu, sigma)

Arguments

Details

The following is the probability density function of the logistic distribution:

f(x)=\frac{e^{-\frac{\left(x-\mu\right)}{\sigma}}}{\sigma\left(1+e^{-\frac{\left(x-\mu\right)}{\sigma}}\right)^{2}},

where x\in\left(-\infty,+\infty\right), \mu\in\left(-\infty,+\infty\right) and \sigma > 0.

Value

d_logis gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the logistic distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, Volume 2 (Vol. 289). John Wiley & Sons.

See Also

d_lnormal

Examples

d_logis(4,0.2)

Compute the distributional properties of the Lomax distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Lomax distribution.

Usage

d_lom(alpha, beta)

Arguments

Details

The following is the probability density function of the Lomax distribution:

f(x)=\frac{\alpha}{\beta}\left(1+\frac{x}{\beta}\right)^{-\alpha-1},

where x > 0, \alpha > 0 and \beta > 0.

Value

d_lom gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Lomax distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Abd-Elfattah, A. M., Alaboud, F. M., & Alharby, A. H. (2007). On sample size estimation for Lomax distribution. Australian Journal of Basic and Applied Sciences, 1(4), 373-378.

See Also

d_gamma

Examples

d_lom(10,10)

Compute the distributional properties of the Nakagami distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Nakagami distribution.

Usage

d_naka(alpha, beta)

Arguments

Details

The following is the probability density function of the Nakagami distribution:

f(x)=\frac{2\alpha^{\alpha}}{\Gamma(\alpha)\beta^{\alpha}}x^{2\alpha-1}e^{-\frac{\alpha x^{2}}{\beta}},

where x > 0, \alpha > 0 and \beta > 0.

Value

d_naka gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Nakagami distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Schwartz, J., Godwin, R. T., & Giles, D. E. (2013). Improved maximum-likelihood estimation of the shape parameter in the Nakagami distribution. Journal of Statistical Computation and Simulation, 83(3), 434-445.

See Also

d_gamma

Examples

d_naka(2,2)

Compute the distributional properties of the normal distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the normal distribution.

Usage

d_normal(alpha, beta)

Arguments

Details

The following is the probability density function of the normal distribution:

f(x)=\frac{1}{\beta\sqrt{2\pi}}e^{-0.5\left(\frac{x-\alpha}{\beta}\right)^{2}},

where x\in\left(-\infty,+\infty\right), \alpha\in\left(-\infty,+\infty\right) and \beta > 0. The parameters \alpha and \beta represent the mean and standard deviation, respectively.

Value

d_normal gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the normal distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Patel, J. K., & Read, C. B. (1996). Handbook of the normal distribution (Vol. 150). CRC Press.

See Also

d_lnormal

Examples

d_normal(4,0.2)

Compute the distributional properties of the Rayleigh distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Rayleigh distribution.

Usage

d_rayl(alpha)

Arguments

Details

The following is the probability density function of the Rayleigh distribution:

f(x)=\frac{x}{\alpha^{2}}e^{-\frac{x^{2}}{2\alpha^{2}}},

where x > 0, \alpha > 0.

Value

d_rayl gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Rayleigh distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Forbes, C., Evans, M. Hastings, N., & Peacock, B. (2011). Statistical Distributions. John Wiley & Sons.

See Also

d_wei

Examples

d_rayl(2)

Compute the distributional properties of the Student distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Student t distribution.

Usage

d_st(v)

Arguments

Details

The following is the probability density function of the Student t distribution:

f(x)=\frac{\Gamma(\frac{v+1}{2})}{\sqrt{v\pi}\Gamma(\frac{v}{2})}\left(1+\frac{x^{2}}{v}\right)^{-(v+1)/2},

where x\in\left(-\infty,+\infty\right) and v > 0.

Value

d_st gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Student t distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Yang, Z., Fang, K. T., & Kotz, S. (2007). On the Student's t-distribution and the t-statistic. Journal of Multivariate Analysis, 98(6), 1293-1304.

Ahsanullah, M., Kibria, B. G., & Shakil, M. (2014). Normal and Student's t distributions and their applications (Vol. 4). Paris, France: Atlantis Press.

See Also

d_chi

Examples

d_st(6)

Compute the distributional properties of the Weibull distribution

Description

Compute the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Weibull distribution.

Usage

d_wei(alpha, beta)

Arguments

Details

The following is the probability density function of the Weibull distribution:

f(x)=\frac{\beta}{\alpha}\left(\frac{x}{\alpha}\right)^{\beta-1}e^{-(\frac{x}{\alpha})^{\beta}},

where x > 0, \alpha > 0 and \beta > 0.

Value

d_wei gives the first four ordinary moments, central moments, mean, and variance, Pearson's coefficient of skewness and kurtosis, coefficient of variation, median and quartile deviation based on the selected parametric values of the Weibull distribution.

Author(s)

Muhammad Imran.

R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com.

References

Hallinan Jr, Arthur J. (1993). A review of the Weibull distribution. Journal of Quality Technology, 25(2), 85-93.

See Also

d_EE

Examples

d_wei(2,2)