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# Gamma & Beta

### Input.

Input for the gamma function value in the top 'X' box.

Input for the gamma distribution parameters in the 'c' and 'b' box. The Erlang distribution is a special case of the Gamma distribution were one value, in the c-box, has to be an integer.

Input for the incomplete gamma function G(c,X): enter an integer in the 'c' box and an integer or a real in the X box. Non-integers in the c box are truncated.

For the cummulative probability and the density value for the gamma distribution fill in the c and b box and additionally the Expectation box. All three are values between zero and infinity. For the cummulative Gamma value only the Erlang distribution is used. This requires the parameter in the c box to be an integer. Non integers are truncated.

To revert the cummulative gamma distribution fill in the c and b box and additionally in the % box a value between 0 and 1. For the cummulative Gamma value only the Erlang distribution is used. This requires the parameter in the c box to be an integer. Non integers are truncated.

Input for the beta function value and the beta distribution parameters in the p and the q boxes.

Input for incomplete beta value, the cummulative probability of the Beta distribution. A value between 'zero' and 'one' in the top 'E' box, an integer value in the second 'p' box and a real or integer value in the third, 'q' box.

To revert the cummulative beta distribution fill in the c and b box and additionally in the % box a value between 0 and 1.

### Discussion.

The procedures are specialist.

Gamma function has an integer and a real procedure. The real procedure is based on Gautschi's CACM 221 and provides 10 digit precision (in principle) up to the value 5, four digits up to 6, one digit up to 7 and no precision beyond 7. The integer procedure works with factorials and is exact, as far as computers are exact. The Erlang distribution is a special case of the Gamma distribution were one value, in the c-box, has to be an integer.

The incomplete gamma function is based on the Poisson and is exact. The way the Poisson is implemented in SISA allows for the X parameter to be a real and still get an exact result.

The complete beta function is based on the Gamma value and the same considerations apply. There is a log-Beta for in case the sum of p and q is large.

The incomplete beta function is based on the Binomial and is exact. The way the Binomial is implemented in SISA allows for the q parameter to be a real and still get an exact result.

Please study the beta and gamma distributions further by using the beta and gamma spreadsheets

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