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# ELPOISSON Procedure

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

### Option

 `BOUND` = scalar Boundary of lower part of distribution

### Parameters

 `MEANS` = variates or scalars Means of the distributions `EXPECTEDVALUES` = variates or scalars Saves the expected values `CLPROBABILITIES` = variates or scalars Saves the cumulative lower probabilities

### Description

`ELPOISSON` calculates expected values of the lower parts of Poisson distributions. The calculation is for all values less 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 lower probabilities can similarly be saved by the `CLPROBABILITIES` parameter.

An important use of `ELPOISSON` 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 one to `BOUND`, of

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

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 lower probability.

Directive: `DISTRIBUTION`.
Procedures: `EUPOISSON``RTOBITPOISSON``GLTOBITPOISSON``HGTOBITPOISSON`.
Commands for: Basic and nonparametric statistics.

### Example

```CAPTION   'ELPOISSON example',\
'Expected values for means 5-25 with a lower bound of 10.';\
STYLE=meta,plain
VARIATE   [VALUES=5...25] Means
ELPOISSON [BOUND=10] Means; EXPECTEDVALUES=Expected; CLPROBABILITIES=Clpr
PRINT     Means,Expected,Clpr; DECIMALS=0,3,4```
Updated on May 26, 2023