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

r_pois {DPQ}

R Documentation

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

Description

Compute

r_v(i) := (v^i / i!) / e_{i-1}(v),

where v =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 v and/or large i?
For which i ( := i_m(v)) is it maximal?
When does r_{λ}(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_v(i) values.
r_pois_expr(): an expression.

Author(s)

Martin Maechler, 20 Jan 2004

Examples

plRpois(12)
plRpois(120)

[Package DPQ version 0.5-2 Index]