Calculates expected values of the upper parts of Poisson distributions (R.W. Payne).

### Option

`BOUND` = scalar |
Boundary of upper part of distribution |

### Parameters

`MEANS` = variates or scalars |
Means of the distributions |

`EXPECTEDVALUES` = variates or scalars |
Saves the expected values |

`CUPROBABILITIES` = variates or scalars |
Saves the cumulative upper probabilities |

### Description

`EUPOISSON`

calculates expected values of the upper parts of Poisson distributions. The calculation is for all values greater than or equal to the value specified by the `BOUND`

option. The `MEANS`

parameter specifies a variate containing means of the distributions, or it can be set to a scalar if there is a single distribution. The expected values can be saved in either a variate or a scalar (to match the type of `MEANS`

) by the `EXPECTEDVALUES`

parameter. The corresponding cumulative upper probabilities can similarly be saved by the `CUPROBABILITIES`

parameter.

An important use of `EUPOISSON`

is to provide estimates for censored Poisson observations for the `RTOBITPOISSON`

, `GLTOBITPOISSON`

and `HGTOBITPOISSON`

procedures. Option: `BOUND`

.

Parameters: `MEANS`

, `EXPECTEDVALUES`

, `CLPROBABILITIES`

.

### Method

The expected value for each mean is given by the sum, for `n`

running from `BOUND`

to `nmax`

, of

`n * PRPOISSON(n; mean) / CUPOISSON(BOUND-1; mean)`

The upper limit `nmax`

is taken as the highest value for which the Poisson probability is greater than or equal to 1.0 E-14. For means up to 1300, this is calculated by a quartic polynomial. A Normal approximation is used the values greater than that.

Problems arise when the cumulative upper probability becomes very low. The calculation becomes unreliable, due to numerical round-off, for values less than 1.0 E-10. A missing value is then returned for the expected value, and a zero value for the cumulative upper probability.

### See also

Directive: `DISTRIBUTION`

.

Procedures: `ELPOISSON`

, `RTOBITPOISSON`

, `GLTOBITPOISSON`

, `HGTOBITPOISSON`

.

Commands for: Basic and nonparametric statistics.

### Example

CAPTION 'EUPOISSON example',\ 'Expected values for means 101-120 with a lower bound of 100.';\ STYLE=meta,plain VARIATE [VALUES=101...120] Means EUPOISSON [BOUND=100] Means; EXPECTEDVALUES=Expected; CUPROBABILITIES=Cupr PRINT Means,Expected,Cupr; DECIMALS=0,3,4