Poisson Distribution

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