r_pois {DPQ} | R Documentation |
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”.
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")
i |
integer .. |
lambda |
non-negative number ... |
iset |
..... |
do.main |
|
type |
type of (line) plot, see |
log |
string specifying if (and where) logarithmic scales should be
used, see |
cex |
character expansion factor. |
col |
colors for the two curves. |
do.eaxis |
|
sub10 |
argument for |
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
.
r_pois()
returns a numeric vector r_v(i) values.
r_pois_expr()
an expression
.
Martin Maechler, 20 Jan 2004
dpois()
.
plRpois(12) plRpois(120)