Poisson distribution with rate parameter $$\lambda$$.

expValPois(lambda)

varPois(lambda)

expValTruncPois(d, lambda, k0, less.than.d = TRUE)

TVatRPois(kap, lambda, k0)

mgfPois(t, lambda)

pgfPois(t, lambda)

## Arguments

lambda

Rate parameter $$\lambda$$.

d

cut-off value.

k0

point up to which to sum the distribution for the approximation.

less.than.d

logical; if TRUE (default) truncated mean for values <= d, otherwise, for values > d.

kap

probability.

t

t.

## Value

Function :

• expValPois gives the expected value.

• varPois gives the variance.

• expValTruncPois gives the truncated mean.

• TVatRPois gives the Tail Value-at-Risk.

• mgfPois gives the moment generating function (MGF).

• pgfPois gives the probability generating function (PGF).

Invalid parameter values will return an error detailing which parameter is problematic.

## Details

The Poisson distribution with rate parameter $$\lambda$$ has probability mass function : $$Pr(X = k) = \frac{\lambda^k \textrm{e}^{-\lambda}}{k!}$$ for $$k = 0, 1, 2, \dots$$, and $$\lambda > 0$$

## Examples

expValPois(lambda = 3)
#> [1] 3

varPois(lambda = 3)
#> [1] 3

expValTruncPois(d = 0, lambda = 2, k0 = 2E2, less.than.d = FALSE)
#> This is an approximation
#> [1] 2
expValTruncPois(d = 2, lambda = 2, k0 = 2E2, less.than.d = TRUE)
#> This is an approximation
#> [1] 0.8120117

TVatRPois(kap = 0.8, lambda = 3, k0 = 2E2)
#> This is an approximation
#> [1] 5.596787

mgfPois(t = 1, lambda = 3)
#> [1] 173.269

pgfPois(t = 1, lambda = 3)
#> [1] 1