# rne

rne(x)
Distribution bounded geometric
Mean approx. $x/(x-1)$
Standard deviation approx. $\sqrt{x}/(x-1)$

rne returns a random number with a bounded geometric distribution. That is, each possible return greater than 1 has a probability that is a fixed fraction of the next lower return, up to a given limit. The fraction is 1 divided by the parameter of rne; thus rne(3) returns 2 one third as often as it returns 1, 3 one third as often as 2, and so on. The maximum return is 5 while the hero's experience level is less than 18; from that point, it is the experience level divided by 3 and rounded down. As experience level cannot exceed 30, the upper bound of rne can never be greater than 10.

A call of rne(3) is used to determine the enchantment of randomly generated weapons, armor, and rings. A call of rne(4) is a component of rnz.

## Mathematical analysis

The effect of experience level on the return from rne is often overstated. Only possible returns greater than 5 are affected, and these are improbable events in any case. Thus, while a level 30 hero could in principle find a random weapon with +10 enchantment, this is a rare event indeed.

Here are the probabilities of each return from rne(3) for experience levels 1 and 30:

Return Level 1 Level 30
1 2/3 2/3
2 2/9 2/9
3 2/27 2/27
4 2/81 2/81
5 1/81 2/243
6 0 2/729
7 0 2/2187
8 0 2/6561
9 0 2/19683
10 0 1/19683

### Description in terms of the Wikipedia article

For the formulae in the Wikipedia article on the geometric distribution, the value of p for a call of rne(x) is the probability that the returned value will be 1; this value is $1-1/x$. The mean and standard deviation for the above infobox are calculated accordingly; but they are approximate, as for the sake of simplicity they give the values for the unbounded geometric distribution, and the return value from rne is bounded.

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