r_pois

0th

Percentile

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

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”.

Keywords
distribution
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 chisquared; in particular ...........

plRpois() simply produces a “nice” plot of r_pois(ii, *) vs ii.

Value

r_pois()

returns a numeric vector \(r_\lambda(i)\) values.

r_pois_expr()

an expression.

See Also

dpois().

Aliases
  • r_pois
  • r_pois_expr
  • plRpois
Examples
# NOT RUN {
<!-- %%>> more [incl other plots] in ==>> ../tests/chisq-nonc-ex.R <<== -->
# }
# NOT RUN {
plRpois(12)
plRpois(120)
# }
Documentation reproduced from package DPQ, version 0.3-3, License: GPL (>= 2)

Community examples

Looks like there are no examples yet.