DiscreteDists (version 1.0.0)

dDMOLBE: The DMOLBE distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Discrete Marshall–Olkin Length Biased Exponential DMOLBE distribution with parameters \(\mu\) and \(\sigma\).

Usage

dDMOLBE(x, mu = 1, sigma = 1, log = FALSE)
pDMOLBE(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rDMOLBE(n, mu = 1, sigma = 1)
qDMOLBE(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

Value

dDMOLBE gives the density, pDMOLBE gives the distribution function, qDMOLBE gives the quantile function, rDMOLBE

generates random deviates.

Arguments

x, q

vector of (non-negative integer) quantiles.

mu

vector of the mu parameter.

sigma

vector of the sigma parameter.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are \(P[X <= x]\), otherwise, \(P[X > x]\).

n

number of random values to return.

p

vector of probabilities.

Author

Olga Usuga,

Details

The DMOLBE distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and mass function given by

\(f(x | \mu, \sigma) = \frac{\sigma ((1+x/\mu)\exp(-x/\mu)-(1+(x+1)/\mu)\exp(-(x+1)/\mu))}{(1-(1-\sigma)(1+x/\mu)\exp(-x/\mu)) ((1-(1-\sigma)(1+(x+1)/\mu)\exp(-(x+1)/\mu))}\)

with \(\mu > 0\) and \(\sigma > 0\)

References

Aljohani2023DiscreteDists

See Also

DMOLBE.

Examples

# Example 1
# Plotting the mass function for different parameter values

x_max <- 20
probs1 <- dDMOLBE(x=0:x_max, mu=0.5, sigma=0.5)
probs2 <- dDMOLBE(x=0:x_max, mu=5, sigma=0.5)
probs3 <- dDMOLBE(x=0:x_max, mu=1, sigma=2)

# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
     ylab="P(X=x)", xlab="X", main="Probability for DMOLBE",
     ylim=c(0, 0.80))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
       legend=c("mu=0.5, sigma=0.5",
                "mu=5, sigma=0.5",
                "mu=1, sigma=2"))

# Example 2
# Checking if the cumulative curves converge to 1

x_max <- 20
cumulative_probs1 <- pDMOLBE(q=0:x_max, mu=0.5, sigma=0.5)
cumulative_probs2 <- pDMOLBE(q=0:x_max, mu=5, sigma=0.5)
cumulative_probs3 <- pDMOLBE(q=0:x_max, mu=1, sigma=2)

plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
     type="o", las=1, ylim=c(0, 1),
     main="Cumulative probability for DMOLBE",
     xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
       legend=c("mu=0.5, sigma=0.5",
                "mu=5, sigma=0.5",
                "mu=1, sigma=2"))

# Example 3
# Comparing the random generator output with
# the theoretical probabilities

x_max <- 15
probs1 <- dDMOLBE(x=0:x_max, mu=5, sigma=0.5)
names(probs1) <- 0:x_max

x <- rDMOLBE(n=1000, mu=5, sigma=0.5)
probs2 <- prop.table(table(x))

cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
nombres <- cn
mp <- barplot(height, beside = TRUE, names.arg = nombres,
              col=c('dodgerblue3','firebrick3'), las=1,
              xlab='X', ylab='Proportion')
legend('topright',
       legend=c('Theoretical', 'Simulated'),
       bty='n', lwd=3,
       col=c('dodgerblue3','firebrick3'), lty=1)

# Example 4
# Checking the quantile function

mu <- 3
sigma <-3
p <- seq(from=0, to=1, by=0.01)
qxx <- qDMOLBE(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
     main="Quantiles of DMOLBE(mu = 3, sigma = 3)")