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”.
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")
r_pois()returns a numeric vector \(r_\lambda(i)\) values.
r_pois_expr()an expression.
integer ..
non-negative number ...
.....
logical specifying if a main
title should be drawn via (main = r_pois_expr).
type of (line) plot, see lines.
string specifying if (and where) logarithmic scales should be
used, see plot.default().
character expansion factor.
colors for the two curves.
logical specifying if
eaxis() (package sfsmisc) should
be used.
argument for eaxis() (with a
different default than the original).
Martin Maechler, 20 Jan 2004
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.
dpois().