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")
integer ..
non-negative number ...
.....
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.
argument for eaxis()
(with a
different default than the original).
r_pois()
returns a numeric vector \(r_\lambda(i)\) values.
r_pois_expr()
an expression
.
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()
.
# NOT RUN {
<!-- %%>> more [incl other plots] in ==>> ../tests/chisq-nonc-ex.R <<== -->
# }
# NOT RUN {
plRpois(12)
plRpois(120)
# }
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