Exponential Distribution

The Exponential distribution is a continuous probability distribution often used to model the time between independent events that occur at a constant average rate. It is defined by the rate parameter $\lambda > 0$.

Usage

Exp(rate = 1)

# S4 method for class 'Exp,numeric'
d(distr, x, log = FALSE)

# S4 method for class 'Exp,numeric'
p(distr, q, lower.tail = TRUE, log.p = FALSE)

# S4 method for class 'Exp,numeric'
qn(distr, p, lower.tail = TRUE, log.p = FALSE)

# S4 method for class 'Exp,numeric'
r(distr, n)

# S4 method for class 'Exp'
mean(x)

# S4 method for class 'Exp'
median(x)

# S4 method for class 'Exp'
mode(x)

# S4 method for class 'Exp'
var(x)

# S4 method for class 'Exp'
sd(x)

# S4 method for class 'Exp'
skew(x)

# S4 method for class 'Exp'
kurt(x)

# S4 method for class 'Exp'
entro(x)

# S4 method for class 'Exp'
finf(x)

llexp(x, rate)

# S4 method for class 'Exp,numeric'
ll(distr, x)

eexp(x, type = "mle", ...)

# S4 method for class 'Exp,numeric'
mle(distr, x, na.rm = FALSE)

# S4 method for class 'Exp,numeric'
me(distr, x, na.rm = FALSE)

vexp(rate, type = "mle")

# S4 method for class 'Exp'
avar_mle(distr)

# S4 method for class 'Exp'
avar_me(distr)

Arguments

rate: numeric. The distribution parameter.
distr: an object of class Exp.
x: For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Exp. For the log-likelihood and the estimation functions, x is the sample of observations.
log, log.p: logical. Should the logarithm of the probability be returned?
q: numeric. Vector of quantiles.
lower.tail: logical. If TRUE (default), probabilities are $P(X \leq x)$, otherwise $P(X > x)$.
p: numeric. Vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.
type: character, case ignored. The estimator type (mle or me).
...: extra arguments.
na.rm: logical. Should the NA values be removed?

Value

Each type of function returns a different type of object:

Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.
Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.
Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.
Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

Details

The probability density function (PDF) of the Exponential distribution is given by: $$ f(x; \lambda) = \lambda e^{-\lambda x}, \quad x \geq 0 .$$

Examples

# -----------------------------------------------------
# Exp Distribution Example
# -----------------------------------------------------

# Create the distribution
rate <- 5
D <- Exp(rate)

# ------------------
# dpqr Functions
# ------------------

d(D, c(0.3, 2, 10)) # density function
#> [1] 1.115651e+00 2.269996e-04 9.643749e-22
p(D, c(0.3, 2, 10)) # distribution function
#> [1] 0.7768698 0.9999546 1.0000000
qn(D, c(0.4, 0.8)) # inverse distribution function
#> [1] 0.1021651 0.3218876
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself
#>   [1] 1.89987690 2.36931267 2.87148233 2.74866636 1.81625942 2.14904160
#>   [7] 0.28175009 4.56760768 1.09349174 0.55779297 4.35602554 0.18890265
#>  [13] 4.62891545 3.88241279 3.74935436 4.79008384 0.02844234 2.61570046
#>  [19] 3.53960260 3.63514303 1.85420176 0.34136504 2.15088222 0.04922910
#>  [25] 4.10220503 1.70539679 4.23510894 0.51545035 0.32568537 4.26716539
#>  [31] 1.82615671 1.48345821 2.23857336 3.83841219 2.00453620 2.29161131
#>  [37] 1.51727683 1.46398022 2.60290035 1.53101548 2.67608246 1.41405854
#>  [43] 2.19931336 4.18380765 2.90573574 0.28884700 3.86703149 2.86690371
#>  [49] 2.07474056 2.81270867 3.62690085 0.83397951 1.70165990 3.94251708
#>  [55] 1.19258253 4.01383551 0.62372620 2.43040021 0.40834374 4.10935165
#>  [61] 1.22289564 3.74197009 1.35856446 3.50883604 1.55563954 0.71899625
#>  [67] 1.36363264 3.03567438 3.50887986 2.24864180 1.86893107 1.79055463
#>  [73] 1.23685773 4.31248100 2.69409546 1.32429995 0.90656930 0.51632727
#>  [79] 2.01382100 2.04640433 0.81883860 4.77134465 1.69830222 2.57596949
#>  [85] 2.50862141 2.49922833 1.24203905 4.40907816 4.78664006 3.93452419
#>  [91] 2.90532206 3.43348595 1.47234436 0.26083641 4.66876037 4.80286053
#>  [97] 3.07056132 3.51662543 2.99126895 1.80387736

# ------------------
# Moments
# ------------------

mean(D) # Expectation
#> [1] 0.2
median(D) # Median
#> [1] 0.1386294
mode(D) # Mode
#> [1] 0
var(D) # Variance
#> [1] 0.04
sd(D) # Standard Deviation
#> [1] 0.2
skew(D) # Skewness
#> [1] 2
kurt(D) # Excess Kurtosis
#> [1] 6
entro(D) # Entropy
#> [1] -0.6094379
finf(D) # Fisher Information Matrix
#> [1] 0.04

# List of all available moments
mom <- moments(D)
mom$mean # expectation
#> [1] 0.2

# ------------------
# Point Estimation
# ------------------

ll(D, x)
#> [1] 59.8082
llexp(x, rate)
#> [1] 59.8082

eexp(x, type = "mle")
#> $rate
#> [1] 4.943858
#> 
eexp(x, type = "me")
#> $rate
#> [1] 4.943858
#> 

mle(D, x)
#> $rate
#> [1] 4.943858
#> 
me(D, x)
#> $rate
#> [1] 4.943858
#> 
e(D, x, type = "mle")
#> $rate
#> [1] 4.943858
#> 

mle("exp", x) # the distr argument can be a character
#> $rate
#> [1] 4.943858
#> 

# ------------------
# Estimator Variance
# ------------------

vexp(rate, type = "mle")
#> rate 
#>   25 
vexp(rate, type = "me")
#> rate 
#>   25 

avar_mle(D)
#> rate 
#>   25 
avar_me(D)
#> rate 
#>   25 

v(D, type = "mle")
#> rate 
#>   25

Usage

Arguments

Value

Details

See also

Examples