distributionsrd (version 0.0.6)

exp: The Exponential distribution

Description

Raw moments for the exponential distribution.

Usage

mexp(r = 0, truncation = 0, rate = 1, lower.tail = TRUE)

Arguments

r

rth raw moment of the distribution, defaults to 1.

truncation

lower truncation parameter, defaults to 0.

rate

rate of the distribution with default values of 1.

lower.tail

logical; if TRUE (default), moments are \(E[x^r|X \le y]\), otherwise, \(E[x^r|X > y]\)

Value

Returns the truncated rth raw moment of the distribution.

Details

Probability and Cumulative Distribution Function:

$$f(x) = \frac{1}{s}e^{-\frac{\omega}{s}} , \qquad F_X(x) = 1-e^{-\frac{\omega}{s}}$$

The y-bounded r-th raw moment of the distribution equals:

$$s^{\sigma_s - 1} \Gamma\left(\sigma_s +1, \frac{y}{s} \right)$$

where \(\Gamma(,)\) denotes the upper incomplete gamma function.

Examples

# NOT RUN {
## The zeroth truncated moment is equivalent to the probability function
pexp(2, rate = 1)
mexp(truncation = 2)

## The (truncated) first moment is equivalent to the mean of a (truncated) random sample,
#for large enough samples.
x <- rexp(1e5, rate = 1)
mean(x)
mexp(r = 1, lower.tail = FALSE)

sum(x[x > quantile(x, 0.1)]) / length(x)
mexp(r = 1, truncation = quantile(x, 0.1), lower.tail = FALSE)
# }