The Exponential Distribution — nv

Density (nv_dexp), distribution function (nv_pexp), and quantile function (nv_qexp) for the Exponential distribution with rate rate (i.e., mean 1 / rate).

Usage

nv_dexp(x, rate = 1, log = FALSE)

nv_pexp(q, rate = 1, lower_tail = TRUE, log_p = FALSE)

nv_qexp(p, rate = 1, lower_tail = TRUE, log_p = FALSE)

Arguments

x, q: (arrayish)
Quantiles at which to evaluate the density (x) or the distribution function (q).
rate: (arrayish)
Rate of the distribution. Scalars broadcast; otherwise rate and x/q/p must have the same shape. Negative rates (including -Inf and negative zero) give NaN.
log, log_p: (logical(1))
If TRUE, the densities/probabilities are given as logarithms. For nv_qexp this describes the input p.
lower_tail: (logical(1))
If TRUE (default), probabilities are $P(X \le x)$; otherwise, $P(X > x)$.
p: (arrayish)
Probabilities at which to evaluate the quantile function. Values outside $[0, 1]$ give NaN. With log_p = TRUE, supply log-probabilities in $[-\infty, 0]$ instead.

Value

(arrayish)
nv_dexp(), nv_pexp(), and nv_qexp() return an arrayish with the float data type described under Data Types and the shape after scalar broadcasting with rate.

Details

The Exponential distribution has probability density function: $$f(x) = \lambda e^{-\lambda x}, \quad x \ge 0$$ and zero elsewhere, where $\lambda$ is rate.

For positive finite rates, the mean is 1 / rate. Zero and infinite rates follow base R's endpoint conventions:

rate = 0: the density and lower-tail probability are zero at finite quantiles, but NaN at infinity. Quantiles are infinite except at lower-tail probability zero, where the result is zero.
rate = Inf: densities are NaN. The lower-tail probability is zero at or below zero and one above zero. Quantiles are zero except at lower-tail probability one, where the result is NaN.

Data Types

nv_dexp(), nv_pexp() and nv_qexp() compute at a float data type and return it. If x/q/p is a float, they compute at its data type; an R double settles on the default float. An integer x/q/p (an array or an R integer) is converted to the widest float among the arrays given for rate, or to the default float if there is none. Integers beyond $2^{24}$ are not all exact at f32, so a large integer may change in that conversion.

An R value for rate takes the computing data type. An array is promoted to it, so a wider one (e.g. f64 for an f32 x) is an error.

Examples

x <- nv_array(c(-1, 0, 0.5, 1, 2))
nv_dexp(x)
#> AnvlArray
#>  0.0000
#>  1.0000
#>  0.6065
#>  0.3679
#>  0.1353
#> [ CPUf32{5} ] 
nv_dexp(x, rate = 2)
#> AnvlArray
#>  0.0000
#>  2.0000
#>  0.7358
#>  0.2707
#>  0.0366
#> [ CPUf32{5} ] 
nv_dexp(x, log = TRUE)
#> AnvlArray
#>     -inf
#>   0.0000
#>  -0.5000
#>  -1.0000
#>  -2.0000
#> [ CPUf32{5} ] 

nv_pexp(x)
#> AnvlArray
#>  0.0000
#>  0.0000
#>  0.3935
#>  0.6321
#>  0.8647
#> [ CPUf32{5} ] 
nv_pexp(x, rate = 2)
#> AnvlArray
#>  0.0000
#>  0.0000
#>  0.6321
#>  0.8647
#>  0.9817
#> [ CPUf32{5} ] 
nv_pexp(x, lower_tail = FALSE)
#> AnvlArray
#>  1.0000
#>  1.0000
#>  0.6065
#>  0.3679
#>  0.1353
#> [ CPUf32{5} ] 
nv_pexp(x, log_p = TRUE)
#> AnvlArray
#>     -inf
#>     -inf
#>  -0.9328
#>  -0.4587
#>  -0.1454
#> [ CPUf32{5} ] 

p <- nv_array(c(0.025, 0.5, 0.975))
nv_qexp(p)
#> AnvlArray
#>  0.0253
#>  0.6931
#>  3.6889
#> [ CPUf32{3} ] 
nv_qexp(p, rate = 2)
#> AnvlArray
#>  0.0127
#>  0.3466
#>  1.8444
#> [ CPUf32{3} ] 
nv_qexp(p, lower_tail = FALSE)
#> AnvlArray
#>  3.6889
#>  0.6931
#>  0.0253
#> [ CPUf32{3} ] 
nv_qexp(nv_array(c(-700, -2, -0.1), dtype = "f64"), log_p = TRUE)
#> AnvlArray
#>  9.8597e-305
#>   1.4541e-01
#>   2.3522e+00
#> [ CPUf64{3} ]