R: Compute Relative Size of i-th term of Poisson Distribution...

r_pois {DPQ} | R Documentation

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

### Description

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”.

### 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 chi-squared; in particular ...........

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

### Value

`r_pois()`

  returns a numeric vector r_v(i) values.

`r_pois_expr()`

  an `expression`.

### Author(s)

Martin Maechler, 20 Jan 2004

### See Also

`dpois()`.

### Examples

    plRpois(12)
    plRpois(120)

[Package DPQ version 0.4-4 Index]