r_pois: Compute Relative Size of i-th term of Poisson Distribution...

In DPQ: Density, Probability, Quantile ('DPQ') Computations

Compute Relative Size of i-th term of Poisson Distribution Series

Description

Compute

r_\lambda(i) := (\lambda^i / i!) / e_{i-1}(\lambda),

where \lambda =lambda, and

e_n(x) := 1 + x + x^2/2! + .... + x^n/n!

is the n-th partial sum of \exp(x) = e^x.

Questions: As function of i

• Can this be put in a simple formula, or at least be well approximated for large \lambda and/or large i?

• For which i ( := i_m(\lambda)) is it maximal?

• When does r_{\lambda}(i) become smaller than (f+2i-x)/x = a + b*i ?

NB: This is relevant in computations for non-central chi-squared (and similar non-central distribution functions) defined as weighted sum with “Poisson weights”.

Usage

r_pois(i, lambda)
r_pois_expr  # the R expression() for the asymptotic branch of r_pois()

plRpois(lambda, iset = 1:(2*lambda), do.main = TRUE,
        log = 'xy', type = "o", cex = 0.4, col = c("red","blue"),
        do.eaxis = TRUE, sub10 = "10")

Arguments

i	integer ..
lambda	non-negative number ...
iset	.....
do.main	logical specifying if a main title should be drawn via (main = r_pois_expr).
type	type of (line) plot, see lines.
log	string specifying if (and where) logarithmic scales should be used, see plot.default().
cex	character expansion factor.
col	colors for the two curves.
do.eaxis	logical specifying if eaxis() (package sfsmisc) should be used.
sub10	argument for eaxis() (with a different default than the original).

Details

r_pois() is related to our series expansions and approximations for the non-central chi-squared; in particular ...........

plRpois() simply produces a “nice” plot of r_pois(ii, *) vs ii.

Value

r_pois()

returns a numeric vector r_\lambda(i) values.

r_pois_expr()

an expression.

Author(s)

Martin Maechler, 20 Jan 2004

See Also

dpois().

Examples

plRpois(12)
plRpois(120)

DPQ documentation built on Dec. 5, 2023, 3:05 a.m.