r_pois: 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

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).

Value

r_pois(): returns a numeric vector $r_\lambda(i)$ values.
r_pois_expr(): an expression.

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.

Examples

Run this code

# NOT RUN {
<!-- %%>> more [incl other plots] in ==>> ../tests/chisq-nonc-ex.R <<== -->
# }
# NOT RUN {
plRpois(12)
plRpois(120)
# }