Bernoulli-geometric distribution

These functions define the density, distribution function, quantile function and random generation for the Bernoulli-geometric distribution with parameters \(\mu\) and \(\sigma\).

dBerG(x, mu, sigma, log = FALSE)

pBerG(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)

rBerG(n, mu, sigma)

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

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.

Value

dBerG gives the density, pBerG gives the distribution function, qBerG gives the quantile function, rBerG generates random deviates.

Details

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

\(f(x | \mu, \sigma) = \frac{(1-\mu+\sigma)}{(1+\mu+\sigma)}\) if \(x=0\),

\(f(x | \mu, \sigma) = 4 \mu \frac{(\mu+\sigma-1)^{x-1}}{(\mu+\sigma+1)^{x+1}}\) if \(x=1, 2, ...\),

with \(\mu > 0\), \(\sigma > 0\) and \(\sigma>|\mu-1|\).

References

bourguignon2022DiscreteDists

Author

Hermes Marques, hermes.marques@ufrn.br

Examples

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

x_max <- 20
probs1 <- dBerG(x=0:x_max, mu=0.7, sigma=0.5)
probs2 <- dBerG(x=0:x_max, mu=0.3, sigma=1)
probs3 <- dBerG(x=0:x_max, mu=2, sigma=3)

# 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 BerG",
     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=0.7, sigma=0.5",
                "mu=0.3, sigma=1",
                "mu=2, sigma=3"))


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

#plot1
x_max <- 10
plot_discrete_cdf(x=0:x_max,
                  fx=dBerG(x=0:x_max, mu=1, sigma=2),
                  col="dodgerblue",
                  main="CDF for BerG",
                  lwd=3)
legend("bottomright", legend="mu=1, sigma=2",
       col="dodgerblue", lty=1, lwd=2, cex=0.8)



#plot2
plot_discrete_cdf(x=0:x_max,
                  fx=dBerG(x=0:x_max, mu=3, sigma=3),
                  col="tomato",
                  main="CDF for BerG",
                  lwd=3)
legend("bottomright", legend="mu=3, sigma=3",
       col="tomato", lty=1, lwd=2, cex=0.8)



#plot3
plot_discrete_cdf(x=0:x_max,
                  fx=dBerG(x=0:x_max, mu=5, sigma=5),
                  col="green4",
                  main="CDF for BerG",
                  lwd=3)
legend("bottomright", legend="mu=5, sigma=5",
       col="green4", lty=1, lwd=2, cex=0.8)



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

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

x <- rBerG(n=1000, mu=0.5, sigma=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 <- 1
sigma <- 2
p <- seq(from=0, to=1, by=0.01)
qxx <- qBerG(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 DBerG(mu=1, sigma=2)")