Unit Maxwell-Boltzmann distribution

These functions define the density, distribution function, quantile function and random generation for the Unit Maxwell-Boltzmann distribution with parameter \(\mu\).

dUMB(x, mu = 1, log = FALSE)

pUMB(q, mu = 1, lower.tail = TRUE, log.p = FALSE)

qUMB(p, mu, lower.tail = TRUE, log.p = FALSE)

rUMB(n = 1, mu = 1)

Arguments

x, q

vector of (non-negative integer) quantiles.

mu

vector of the mu 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, \(P[X > x]\).

p

vector of probabilities.

n

number of random values to return.

Details

The Unit Maxwell-Boltzmann distribution with parameter \(\mu\) has a support in \((0, 1)\) and density given by

\(f(x| \mu) = \frac{\sqrt(2/\pi) \log^2(1/x) \exp(-\frac{\log^2(1/x)}{2\mu^2})}{\mu^3 x} \)

for \(0 < x < 1\) and \(\mu > 0\).

References

Biçer, C., Bakouch, H. S., Biçer, H. D., Alomair, G., Hussain, T., y Almohisen, A. (2024). Unit Maxwell-Boltzmann Distribution and Its Application to Concentrations Pollutant Data. Axioms, 13(4), 226.

See also

UMB.

Author

David Villegas Ceballos, david.villegas1@udea.edu.co

Examples

# Example 1
# Plotting the density function for different parameter values
curve(dUMB(x, mu=0.4), from=0, to=1,
      ylim=c(0, 12),
      col="green", las=1, ylab="f(x)")

curve(dUMB(x, mu=1),
      add=TRUE, col="blue1")

curve(dUMB(x, mu=2),
      add=TRUE, col="black")

curve(dUMB(x, mu=7),
      add=TRUE, col="red")

legend("topright",
       col=c("green", "blue1", "black", "red"),
       lty=1, bty="n",
       legend=c("mu=0.4",
                "mu=1",
                "mu=2",
                "mu=7"))


# Example 2
# Checking if the cumulative curves converge to 1
curve(pUMB(x, mu=0.25),
      from=0, to=1, col="green", las=1, ylab="F(x)")

curve(pUMB(x, mu=0.9),
      add=TRUE, col="blue1")

curve(pUMB(x, mu=1.8),
      add=TRUE, col="black")

curve(pUMB(x, mu=2.2),
      add=TRUE, col="red")

legend("bottomright", col=c("green", "blue1", "black", "red"),
       lty=1, bty="n",
       legend=c("mu=0.25",
                "mu=0.9",
                "mu=1.8",
                "mu=2.2"))


# Example 3
# Checking the quantile function
mu <- 2
p <- seq(from=0, to=1, length.out=100)
plot(x=qUMB(p, mu=mu), y=p,
     xlab="Quantile", las=1, ylab="Probability")
curve(pUMB(x, mu=mu), add=TRUE, col="red")


# Example 4
# Comparing the random generator output with
# the theoretical density
x <- rUMB(n=1000, mu=0.5)
hist(x, freq=FALSE)
curve(dUMB(x, mu=0.5),
      col="tomato", add=TRUE, from=0, to=1)