The gamma distribution in its original parameterization

Density, distribution function, quantile function, random generation and hazard function for the gamma original distribution with parameters mu and sigma.

Usage

dGAo(x, mu = 1, sigma = 1, log = FALSE)

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

qGAo(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rGAo(n, mu = 1, sigma = 1)

hGAo(x, mu, sigma)

Arguments

x, q: vector of quantiles.
mu: parameter.
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].
p: vector of probabilities.
n: number of observations.

Value

dGAo gives the density, pGAo gives the distribution function, qGAo gives the quantile function, rGAo generates random deviates and hGAo gives the hazard function.

Details

The gamma original with parameters mu and sigma has density given by

\(f(x|\mu,\sigma) = \frac{x^{\mu-1}e^{-x/\sigma}}{\sigma^\mu \Gamma(\mu)}\)

for \(x>0\), \(\mu>0\) and \(\sigma>0\). The parameter \(\mu\) is the shape parameter and \(\sigma\) is the scale parameter. In this parameterization \(\mu\) is the median of \(X\), \(E(X)=\mu \sigma\) and \(Var(X)=\mu \sigma^2\).

References

Abramowitz M, Stegun IA (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York. ISBN 0486612724. Chapter 6: Gamma and Related Functions.

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dGAo(x, mu=7.5, sigma=1), 
      from=0.001, to=20,
      ylim=c(0, 0.5), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dGAo(x, mu=1, sigma=2),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dGAo(x, mu=5, sigma=1),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=7.5, sigma=1", 
                            "mu=1 , sigma=2",
                            "mu=5, sigma=1"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)



# Example 2
# Checking if the cumulative curves converge to 1
curve(pGAo(x, mu=0.5, sigma=0.5), 
      from=0.001, to=5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="F(x)")
curve(pGAo(x, mu=1, sigma=0.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(pGAo(x, mu=1.5, sigma=0.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=0.5, sigma=0.5", 
                               "mu=1.0, sigma=0.5",
                               "mu=1.5, sigma=0.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.5)



# Example 3
# The quantile function
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qGAo(p, mu=2.3, sigma=1.7), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pGAo(x, mu=2.3, sigma=1.7), 
      from=0, add=TRUE, col="tomato", lwd=2.5)


# Example 4
# The random function
x <- rGAo(n=10000, mu=20, sigma=0.5)
hist(x, freq=FALSE)
curve(dGAo(x, mu=20, sigma=0.5), from=0, to=100, 
      add=TRUE, col="tomato", lwd=2)


# Example 5
# The Hazard function
curve(hGAo(x, mu=20, sigma=0.5), from=0.001, to=100,
      col="tomato", ylab="Hazard function", las=1)