3.9 Special Continuous Distributions

For more details, (Johnson, Kotz, and Balakrishnan 1994) and (Johnson, Kotz, and Balakrishnan 1995) are recommended. McLaughlin (2016) provides a compendium of probability distributions.

3.9.1 Continuous Uniform \(\cdot \; \mathcal{U}(a,b)\)

The continuous uniform distribution in the interval \(\left[ a,b \right]\) has its (probability) density (function) defined by

\[\begin{equation} f(x|a,b) = \dfrac{1}{b-a} \tag{3.72} \end{equation}\]

where \(a \le x \le b\), \(-\infty < a,b < \infty\) with \(b>a\). Cumulative distribution function

\[\begin{equation} F(x|a,b) = Pr(X<x) = \dfrac{x-a}{b-a} \tag{3.73} \end{equation}\]

Expected value \[\begin{equation} E(X) = \dfrac{a+b}{2} \tag{3.74} \end{equation}\]

Variance \[\begin{equation} V(X) = \dfrac{(b-a)^2}{12} \tag{3.75} \end{equation}\]

Exercise 3.21 Demonstrate the Equations (3.73), (3.74) and (3.75) considering the identities (3.76) and (3.77).

\[\begin{equation} a^2 - b^2 = (a-b)(a+b) \tag{3.76} \end{equation}\]

\[\begin{equation} a^3 - b^3 = (a-b)(a^2+ab+b^2) \tag{3.77} \end{equation}\]

3.9.2 Normal \(\cdot \; \mathcal{N}(\mu,\sigma)\)

The normal or Gaussian distribution (named after Johann Carl Friedrich Gauss) is denoted by \(\mathcal{N}(\mu,\sigma)\). Its pdf is given by \[\begin{equation} f(x|\mu,\sigma) = \dfrac{1}{\sqrt{2\pi} \sigma} \exp \bigg\{ -\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right) ^2 \bigg\} \tag{3.78} \end{equation}\]

for \(-\infty < x < \infty\), \(-\infty < \mu < \infty\), \(\sigma > 0\). The parameters \(\mu\) and \(\sigma\) can be calculated respectively by the Equations (2.8) and (2.26). The notation \(\exp\{...\}\) represents the Euler number raised to the expression enclosed by the square brackets.

The standard normal distribution is given by the expression \[\begin{equation} f(z|\mu=0, \sigma=1) = \dfrac{1}{\sqrt{2\pi}} \exp \bigg\{ -\frac{z^2}{2} \bigg\} \tag{3.79} \end{equation}\]

for \(-\infty < z < \infty\).

Example 3.36 The normal distribution can be operated with the functions dnorm (density), pnorm (cumulative probability), qnorm (quantile) and rnorm (random/random) from the stats library.

# Pr(X < 2) in a N(0,1)
pnorm(2)

## [1] 0.9772499

# quantile that limits 2.5% in a N(0,1)
qnorm(0.025)

## [1] -1.959964

# simulating 1000 pseudo-random values of a N(0,1)
set.seed(999); x <- rnorm(1000)
summary(x)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -3.073370 -0.705544 -0.009566 -0.032516  0.636537  3.061137

# density of a N(0,1), overlapping simulated values
hist(x, freq = F,main = 'N(0,1)')
curve(dnorm(x), add = T, col = 'red')

Exercise 3.22 Watch the video But what is the Central Limit Theorem? from the channel 3Blue1Brown. I thank Vitor Luiz Cavagnolli Machado for the suggestion.

Exercise 3.23 Read the article https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule.

Exercise 3.24 Consider the random variable \(Z\) with standard normal distribution denoted by \(Z \sim \mathcal{N}(0,1)\). Obtain the following quantities:
a. \(Pr(Z \le -1.96)\).
b. \(Pr(-1 \le Z \le 1)\).
c. \(Pr(Z > 1.64)\).
d. \(q\) tal que \(Pr(Z \le q) = 0.025\).
e. \(q\) tal que \(Pr(-q \le Z \le q) = 0.6826894921\).
f. \(q\) tal que \(Pr(Z > q) = 0.05\).

Solution: Chapter ??
\(\\\)

Exercise 3.25 Refrigerators produced by a factory have a certain lifespan until the first damage. Studies indicate that this time follows a normal distribution with a mean of 1.45 years and a standard deviation of 0.15 years.

1. The factory offers a 1-year warranty. What is the probability of a refrigerator breaking down during this period?
   
2. How likely is it that a refrigerator will fail outside of warranty?
   
3. What is the probability of a refrigerator failing between the first and second year of use?
   
4. What is the probability of a refrigerator lasting more than 2 years without failing?
   
5. If the factory produced 80 thousand refrigerators, how many people should claim the warranty? \(\\\)

To know more

(Patel and Read 1982) bring a collection of results and properties related to the normal distribution. (Tong 1990) provides a comprehensive treatment of results related to the multivariate normal distribution. The main themes are dependence, probability inequalities and their roles in theory and applications.

3.9.3 Exponential \(\cdot \; \mathcal{E}(\lambda)\)

Consider again the toll described in Section 3.7.4, where on average \(\lambda\) vehicles pass per minute. You can invert the reading, placing the time between each car as the new variable of interest. Thus, 1 car passes through this toll every \(\frac{1}{\lambda}\) minutes. The continuous random variable \(X\): ‘time between vehicles’ has exponential distribution of parameter \(\lambda\), denoted by \[ X \sim \mathcal{E}(\lambda), \] where \(x > 0\) and \(\lambda > 0\) indicates the rate. The exponential density function is given by

\[\begin{equation} f(x|\lambda) = \lambda e^{-\lambda x} \tag{3.80} \end{equation}\] where \(e\) is the Euler’s number. Its cumulative distribution function is given by \[\begin{equation} F(x|\lambda) = Pr(X \le x) = 1 - e^{-\lambda x} \tag{3.81} \end{equation}\] The expected value a nd variance are given by \[\begin{equation} E(X)= \frac{1}{\lambda} = \lambda^{-1} \tag{3.82} \end{equation}\] \[\begin{equation} V(X)=\frac{1}{\lambda^2} = \lambda^{-2} \tag{3.83} \end{equation}\]

Example 3.37 The exponential distribution can be operated with the functions dexp (density), pexp (accumulated probability), qexp (quantile) and rexp (random/random) from the stats library.

# Pr(X < 2) in an exponential of rate 1
pexp(2, 1)

## [1] 0.8646647

# quantile that limits 2.5% in an exponential of rate 1
qexp(0.025, 1)

## [1] 0.02531781

# simulating 1000 pseudo random values of an exponential of rate 1
set.seed(999); x <- rexp(1000, 1)
summary(x)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.00011 0.29347 0.69805 1.00508 1.41511 5.42528

# density of an exponential of rate 1, overlapping simulated values
hist(x, freq = F, ylim = c(0,1), main = 'Exp(1)')
curve(dexp(x,1), add = T, col = 'red')

Example 3.38 Consider a tollbooth where an average of \(\lambda = 2\) vehicles pass per minute. Thus, \[ X \sim \mathcal{E}(2),\] \[ f(x) = 2 e^{-2 x}, \] \[ E(X)=\dfrac{1}{2}=0.5, \] \[ V(X)=\dfrac{1}{2^2}=0.25, \] \[ D(X) = \sqrt{0.25} = 0.5. \]

Exercise 3.26 Considering the data from Example 3.38:
a. Define \(F(x)\).
b. Get \(Pr(X<1)\).
c. Obtain \(Pr(X>2)\).
d. Sketch the graph of \(f(x)\). \(\\\)

Exercise 3.27 Consider the exponential density given by Eq. (3.80), the definitions (3.69), (3.70) and @ref(eq: var-vad-x).
a. Show that \(F(x) = 1-e^{-\lambda x}\).
b. Show that \(E(X) = \lambda^{-1}\).
c. Show that \(V(X) = \lambda^{-2}\). \(\\\)

3.9.4 Student’s \(t\) \(\cdot \; \mathcal{t_\nu}\)

If \(X \sim t_\nu\), then its pdf is given by

\[\begin{equation} f(x|\nu) = \frac{\Gamma \left( \frac{\nu+1}{2} \right)}{\Gamma \left( \frac{\nu}{2} \right) \sqrt{\nu \pi}} \left( 1+\frac{x^2}{\nu} \right)^{-\frac{\nu + 1}{2}} \tag{3.84} \end{equation}\]

where \(-\infty < x < \infty\), \(\nu > 0\) indicates the degrees of freedom and \(\Gamma\) indicates the gamma function such that

\[\begin{equation} \Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt \tag{3.85} \end{equation}\]

Equivalently, one can write down \(X \sim t(\nu)\).

Example 3.39 The \(t\) distribution can be operated with the functions dt (density), pt (accumulated probability), qt (quantile) and rt (random/random) from the stats library.

# Pr(X < 2) in a t with 1 df
pt(2, 1)

## [1] 0.8524164

# quantile that limits 2.5% in a t with 1 df
qt(0.025, 1)

## [1] -12.7062

# simulating 1000 pseudo random values from a t with 1 df
set.seed(246); x <- rt(1000, 1)
summary(x)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -305.1952   -1.1336   -0.0891    0.3420    1.0430 1294.5135

# density of a t with 1 degree of freedom, overlapping simulated values
hist(x, 3000, freq = F, xlim = c(-15,15), main = expression(italic('t')(1)))
curve(dt(x,1), add = T, col = 'red')

3.9.5 Chi-squared \(\cdot \; \mathcal{\chi}^2_\nu\)

If \(X \sim \chi^2_\nu\), then its pdf is given by

\[\begin{equation} f(x|\nu) = \frac{1}{\Gamma \left( \frac{\nu}{2} \right) 2^{\nu/2}} x^{\frac{\nu}{2}-1} e^{-\frac{x}{2}} \tag{3.86} \end{equation}\]

where \(x > 0\), \(\nu > 0\) indicates the degrees of freedom and \(\Gamma\) is the gamma function according to Eq. (3.85). Equivalently, one can write down \(X \sim \chi^2(\nu)\).

Example 3.40 The \(\chi^2\) distribution can be operated with the functions dchisq (density), pchisq (accumulated probability), qchisq (quantile) and rchisq (random/random) from the stats library.

# Pr(X < 2) in a \chi^2 with 1 df
gl <- 1
pchisq(2, gl)

## [1] 0.8427008

# quantile that  limits 2.5% in a \chi^2 with 1 df
qchisq(0.025, gl)

## [1] 0.0009820691

# simulating 1000 pseudo random values from a \chi^2 with 1 df
set.seed(135); x <- rchisq(1000, gl)
summary(x)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.00000 0.08634 0.42728 0.98606 1.30768 8.91451

# density of a \chi^2 with 1 degree of freedom, overlapping simulated values
hist(x, 50, freq = F, main = bquote(chi^2~  (.(gl))))
curve(dchisq(x,1), 0, 10, add = T, col = 'red')

Characterization

\(Q \sim \chi^2_\nu\) if

\[\begin{equation} Q = \sum_{i=1}^\nu Z_i^2 \tag{3.87} \end{equation}\]

where \(Z_i^2 \sim \mathcal{N}(0,1)\), \(i \in \{1,\ldots,\nu\}\).

Example 3.41 It is possible to simulate a chi-square with 1 degree of freedom.

# via standard normal squared
set.seed(1234); q1 <- rnorm(1000)^2
# via rchisq
set.seed(5678); q2 <- rchisq(1000, 1)
par(mfrow=c(1,2))
hist(q1, 30)
hist(q2, 30)

# checking the adherence of the simulations
ks.test(q1, q2)

## 
##  Asymptotic two-sample Kolmogorov-Smirnov test
## 
## data:  q1 and q2
## D = 0.026, p-value = 0.8879
## alternative hypothesis: two-sided

Exercise 3.28 Simulate a chi-square with 3 degrees of freedom via Eq. (3.87). Compare with the simulation obtained via rchisq.

3.9.6 Fisher-Snedecor \(\cdot \; \mathcal{F}_\nu\)

If \(X \sim \mathcal{F}_{\nu_1,\nu_2}\), then its pdf is given by

\[\begin{equation} f(x|\nu_1,\nu_2) = \frac{\sqrt{\frac{(\nu_1 x)^{\nu_1} \nu_2^{\nu_2}}{(\nu_1 x+\nu_2)^{\nu_1 + \nu_2}}}}{x B\left( \frac{\nu_1}{2}, \frac{\nu_2}{2} \right)} \tag{3.88} \end{equation}\]

where \(x > 0\), \(\nu_1 > 0\) indicates the numerator degrees of freedom, \(\nu_2 > 0\) indicates the denominator degrees of freedom and \(B\) indicates the beta function, such that

\[\begin{equation} B(x_1,x_2) = \int_{0}^{1} t^{x_1 - 1} (1-t)^{x_2 - 1} dt = \frac{\Gamma(x_1) \Gamma(x_2)}{\Gamma(x_1+x_2)} \tag{3.89} \end{equation}\]

Example 3.42 The distribution \(\mathcal{F}\) can be operated with the functions df (density), pf (cumulative probability), qf (quantile) and rf (random/random) from the library stats.

# Pr(X < 2) in an F with 1 df in the numerator and 3 df in the denominator
gl1 <- 1
gl2 <- 3
pf(2, gl1, gl2)

## [1] 0.7477845

# quantile that limits 2.5% in an F(1,3)
qf(0.025, gl1, gl2)

## [1] 0.001157189

#simulating 1000 pseudo-random values of an F(1,3)
set.seed(1010); x <- rf(1000, gl1, gl2)
summary(x)

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
##   0.0000   0.1073   0.5564   2.3601   1.9758 120.0880

# density of an F(1,3), overlapping simulated values
hist(x, 500, freq = F, xlim = c(0,10), main = bquote(F(.(gl1),.(gl2))))
curve(df(x, gl1, gl2), 0, 10, add = TRUE, col = 'red')

Characterization

\(X \sim \mathcal{F}_{\nu_1,\nu_2}\) if

\[\begin{equation} X = \frac{Q_1/\nu_1}{Q_2/\nu_2} \tag{3.90} \end{equation}\]

where \(Q_1 \sim \chi^2_{\nu_1}\) and \(Q_2 \sim \chi^2_{\nu_2}\).

Example 3.43 It is possible to simulate a \(\mathcal{F}\) with \(\nu_1=1\) degree of freedom in the numerator and \(\nu_2=3\) degrees of freedom in the denominator.

set.seed(1234); q1 <- rchisq(1000, 1)
set.seed(5678); q2 <- rchisq(1000, 3)
x1 <- (q1/1)/(q2/3)
x2 <- rf(1000, 1, 3)
par(mfrow=c(1,2))
hist(x1, 50, xlim = c(0,20))
hist(x2, 100, xlim = c(0,20))

ks.test(x1, x2)

## 
##  Asymptotic two-sample Kolmogorov-Smirnov test
## 
## data:  x1 and x2
## D = 0.023, p-value = 0.9541
## alternative hypothesis: two-sided

3.9.7 Beta \(\cdot \; \mathcal{Beta}(\alpha,\beta)\)

The beta density function is given by \[\begin{equation} f(x|\alpha,\beta) = \dfrac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1} (1-x)^{\beta-1} \tag{3.91} \end{equation}\] where \(0 \le x \le 1\), \(\alpha,\beta > 0\) and \(\Gamma\) is the gamma function according to Eq. (3.85). Expected value and variance are given by \[\begin{equation} E(X) = \frac{\alpha}{\alpha+\beta} \tag{3.92} \end{equation}\] \[\begin{equation} V(X) = \frac{\alpha\beta}{(\alpha+\beta)^2 (\alpha+\beta+1)} \tag{3.93} \end{equation}\]

The median and mode are given by \[\begin{equation} Md(X) \approx \frac{\alpha-1/3}{\alpha+\beta-2/3} \tag{3.94} \end{equation}\] \[\begin{equation} Mo(X) = \frac{\alpha-1}{\alpha+\beta-2}, \; \alpha,\beta>1 \tag{3.95} \end{equation}\] \[\begin{equation} Mo(X) \; \text{algum valor entre 0 e 1}, \; \alpha=\beta=1 \tag{3.96} \end{equation}\] \[\begin{equation} Mo(X) = \{0,1\}, \; \alpha,\beta<1 \tag{3.97} \end{equation}\] \[\begin{equation} Mo(X) = 0, \; \alpha \le 1,\beta>1 \tag{3.98} \end{equation}\] \[\begin{equation} Mo(X) = 1, \; \alpha>1,\beta \le 1 \tag{3.99} \end{equation}\]

Example 3.44 The Beta distribution can be operated with the functions dbeta (density), pbeta (cumulative probability), qbeta (quantile) and rbeta (random/random) from the stats library.

# Pr(X < 1/2) in a Beta(1,3)
pbeta(1/2, 1, 3)

## [1] 0.875

# quantile that limits 2.5% in a Beta(1,3)
qbeta(0.025, 1, 3)

## [1] 0.008403759

# simulating 1000 pseudo random values of a Beta(1,3)
set.seed(1010); x <- rbeta(1000, 1, 3)
summary(x)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 0.0000842 0.0905449 0.2126896 0.2502407 0.3621014 0.9394182

# density of a Beta(1,3), overlaying simulated values
hist(x, 30, freq = F, ylim = c(0, 3), main = 'Beta(1,3)')
curve(dbeta(x,1,3), add = T, col = 'red')

3.9.8 Gamma \(\cdot \; \mathcal{Gama}(k,g)\)

The gamma density function can be given by \[\begin{equation} f(x|k,g) = \frac{1}{\Gamma(k) g^k} x^{k-1} e^{-\frac{x}{g}} \tag{3.100} \end{equation}\] where \(x>0\), \(k>0\) (shape), \(g>0\) (scale) and \(\Gamma\) is the gamma function according to Eq. (3.85). Expected value and variance are given by \[\begin{equation} E(X) = kg \tag{3.101} \end{equation}\] \[\begin{equation} V(X) = kg^2 \tag{3.102} \end{equation}\]

The median does not have a closed form, and the mode is given by \[\begin{equation} Mo(X) = (k-1)g, \;\; k \ge 1 \tag{3.103} \end{equation}\] \[\begin{equation} Mo(X) = 0, \;\; k < 1 \tag{3.103} \end{equation}\]

Example 3.45 The Gamma distribution can be operated with the functions dgamma (density), pgamma (accumulated probability), qgamma (quantile) and rgamma (random/random) from the stats library.

# Pr(X < 2) in a Gamma(1,3)
pgamma(2, 1, 3)

## [1] 0.9975212

# quantile that limits 2.5% in a Gamma(1,3)
qgamma(0.025, 1, 3)

## [1] 0.008439269

# simulating 1000 pseudo-random values of a Gamma(1,3)
set.seed(1010); x <- rgamma(1000, 1, 3)
summary(x)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 0.0003625 0.0914895 0.2279837 0.3381387 0.4639010 2.6271045

# density of a Gamma(1,3), overlapping simulated values
hist(x, 30, freq = F, ylim = c(0, 3), main = 'Gama(1,3)')
curve(dgamma(x,1,3), add = T, col = 'red')

Exercise 3.29 Consider the following parameterization of the Gamma distribution: \[\begin{equation} f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} \tag{3.104} \end{equation}\] where \(x>0\), \(\alpha>0\) (shape) and \(\beta>0\) (rate).

1. Check that the shape parameter \(k\) of Eq. (3.100) is equal to the shape parameter \(\alpha\) of Eq. (3.104).
2. Check that the scale parameter \(g\) of Eq. (3.100) is the inverse of the rate \(\beta\) of Eq. (3.104).

3.9.9 Triangular \(\cdot \; \mathcal{Tri}(a,m,b)\)

(Samuel Kotz and Van Dorp 2004) define the triangular density function in the interval \([a,b]\) with mode \(m\) by \[\begin{equation} f(x|a,b,m) = \left\{ \begin{array}{l} \frac{2}{b-a} \frac{x-a}{m-a}, \;\; a \le x \le m \\ \frac{2}{b-a} \frac{b-x}{b-m}, \;\; m < x \le b \\ \end{array} \right. \tag{3.105} \end{equation}\]

where \(a \le x \le b\) e \(a \le m \le b\), \(-\infty < a,b < \infty\) with \(b>a\).

Its cumulative distribution function is given by \[\begin{equation} F(x|a,b,m) = Pr(X \le x) = \left\{ \begin{array}{l} \frac{m-a}{b-a} \left( \frac{x-a}{m-a} \right)^2, \;\; a \le x \le m \\ 1 - \frac{b-m}{b-a} \left( \frac{b-x}{b-m} \right)^2, \;\; m < x \le b \\ \end{array} \right. \tag{3.106} \end{equation}\]

Its inverse cumulative distribution function is given by \[\begin{equation} F^{-1}(u|a,b,m) = \left\{ \begin{array}{l} a + \sqrt{u(m-a)(b-a)}, \;\; 0 \le u \le \frac{m-a}{b-a} \\ b - \sqrt{(1-u)(b-m)(b-a)}, \;\; \frac{m-a}{b-a} < u \le 1 \\ \end{array} \right. \tag{3.107} \end{equation}\]

(Millard 2013) presents the EnvStats package, which has varied functions for Environmental Statistics that consider the triangular distribution.

library(EnvStats)
set.seed(2); hist(rtri(10000), 40, freq = FALSE, main = 'Tri(0,1/2,1)')
curve(dtri(x), col = 'red', add = TRUE)

Exercise 3.30 See the documentation for ?EnvStats::dtri.

3.9.10 Gompertz \(\cdot \; \mathcal{Gompertz}(\alpha,\beta)\)

(Gompertz 1825) defines the Gompertz density function of shape parameter \(\alpha>0\) and scale \(\beta>0\) for \(x>0\) by \[\begin{equation} f(x|\alpha,\beta) = \alpha \beta \exp \left\{ \beta x + \alpha - \alpha e^{\beta x} \right\} \tag{3.108} \end{equation}\]

Its cumulative distribution function is given by \[\begin{equation} F(x|\alpha,\beta) = 1 - \exp \left\{ - \alpha(e^{\beta x}-1) \right\} \tag{3.109} \end{equation}\]

(Yee 2010) presents functions for the Gompertz distribution.

library(VGAM)
curve(dgompertz(x, scale = 1, shape = .1), xlim = c(0,5), ylim = c(0,1.2), 
      col = 'red')
curve(dgompertz(x, scale = 1, shape = 2), xlim = c(0,5), ylim = c(0,1.2), 
      col = 'black', add = TRUE)
curve(dgompertz(x, scale = 1, shape = 3), xlim = c(0,5), ylim = c(0,1.2), 
      col = 'blue', add = TRUE)
curve(dgompertz(x, scale = 2, shape = 1), xlim = c(0,5), ylim = c(0,1.2), 
      col = 'green', add = TRUE)

3.9.11 Unit-Gompertz \(\cdot \; \mathcal{GU}(\alpha,\beta)\)

(Mazucheli, Menezes, and Dey 2019) define the Unit-Gompertz distribution from the type transformation \[X=e^{-Y}\]

where \(Y\) has a Gompertz distribution. Its density function with parameters \(\alpha>0\) and \(\beta>0\) with \(0<x<1\) is given by

\[\begin{equation} f(x|\alpha,\beta) = \alpha \beta x^{-(\beta+1)} \exp \left\{ -\alpha(x^{-\beta}-1) \right\} \tag{3.110} \end{equation}\]

Its cumulative distribution function is given by \[\begin{equation} F(x|\alpha,\beta) = \exp \left\{ -\alpha(x^{-\beta}-1) \right\} \tag{3.111} \end{equation}\]

(Menezes and Mazucheli 2021) present the unitquantreg package, that provides a collection of parametric quantile regression models for bounded data. The autors also present functions for the unit-Gompertz distribution reparametrized in terms of the \(\tau\)-th quantile, \(\tau \in (0,1)\).

library(unitquantreg)
set.seed(123)
x <- rugompertz(n = 5000, mu = 0.5, theta = 2, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'Gompertz unit')
lines(S, dugompertz(x = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)

plot(ecdf(x))
lines(S, pugompertz(q = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)

plot(quantile(x, probs = S), type = "l")
lines(qugompertz(p = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)

3.9.12 Continuous Poisson

(Ilienko 2013) presents and discusses continuous counterparts of the Poisson and binomial distributions.

References

Gompertz, Benjamin. 1825. “On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies. In a Letter to Francis Baily, Esq. FRS &c.” Philosophical Transactions of the Royal Society of London, no. 115: 513–83. https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1825.0026.

Ilienko, Andrii. 2013. “Continuous Counterparts of Poisson and Binomial Distributions and Their Properties.” arXiv Preprint arXiv:1303.5990. https://arxiv.org/abs/1303.5990.

Johnson, Norman L, Samuel Kotz, and Narayanaswamy Balakrishnan. 1994. Continuous Univariate Distributions. 2nd ed. John Wiley & Sons.

———. 1995. Continuous Univariate Distributions. 2nd ed. Vol. 2. John Wiley & Sons.

Kotz, Samuel, and Johan René Van Dorp. 2004. Beyond Beta: other continuous families of distributions with bounded support and applications. World Scientific. https://doi.org/10.1142/5720.

Mazucheli, Josmar, André Felipe Menezes, and Sanku Dey. 2019. “Unit-Gompertz Distribution with Applications.” Statistica 79 (1): 25–43. https://rivista-statistica.unibo.it/article/download/8497/9401.

Menezes, Andr’e F. B., and Josmar Mazucheli. 2021. Unitquantreg: Parametric Quantile Regression Models for Bounded Data. https://andrmenezes.github.io/unitquantreg/.

Millard, Steven P. 2013. EnvStats: An R Package for Environmental Statistics. New York: Springer. https://link.springer.com/book/9781461484554.

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