# rnz

rnz(x)

 Probability density function of a call of rnz(100).

Distribution
unique

Mean
approx. 1.29815x

Standard deviation
approx. 1.04424x

rnz is a function that generates a random number with a distribution that is meant to peak at a given value and trail off at larger values.  A call of rnz(350), for example, gives the base prayer timeout and will often return a number close to 350, sometimes a number somewhat less or somewhat more than 350, and rarely a number much greater or much less than 350.
The most probable outcome (the mode of the distribution) turns out not to be the given parameter, but half of that number, as can be seen in the graph to the right.  The mean outcome, on the other hand, is about 1.30 times the given parameter.
rnz is used for the following purposes:

rnz(100) is the time that one must wait after invoking an artifact before invoking it again.
rnz(10) (or rnz(25) when first building a level) plays a part in determining when a corpse disintegrates.
A number of rnz calls are used in determining the prayer timeout.  In particular, the timeout after successful prayer is rnz(350), with a penalty of rnz(1000) added if you are crowned or have killed the Wizard of Yendor.  If you have killed the Wizard and are also crowned, the penalty is doubled.
Contents

1 Mathematical analysis

1.1 Probability density function
1.2 Cumulative distribution function
1.3 Mean
1.4 Standard deviation

Mathematical analysis
rnz produces such a bizarre distribution that it is hard to tell what the original programmer had in mind.  It's quite possible that it was meant to be some distribution with Z in the name, but its construction doesn't seem to suggest one.
The code is quoted below, with some cleanup for readability:

int
rnz(i)
int i;
{
        register long x = i;
        register long tmp = 1000;

        tmp += rn2(1000);
        tmp *= rne(4);
        if (rn2(2)) { x *= tmp; x /= 1000; }
        else { x *= 1000; x /= tmp; }
        return((int)x);
}

Probability density function
The rn2(2) call chooses, with equal probability, between two different distributions; one is

and the other is

.
The function is easier to analyze if the expression

is modeled as a continuous uniformly distributed variable with range from 1 to 2.  Let this variable be called z.  For the case where the variable i is divided by the random formula, it helps to know the distribution of the reciprocal of z; it is z-2 for 0.5 &lt;= z &lt;= 1, and 0 otherwise.
The rne(4) call returns an integer from 1 to 5, with the following probabilities:

Return
Probability

1
3/4

2
3/16

3
3/64

4
3/256

5
1/256

If the hero's experience level is 18 or greater, then rne can return numbers greater than 5; but this event has low probability (1/1024 for all experience levels 18 or greater), and to keep this explanation simple it will not be considered.
We can now consider all possible outcomes of the rn2(2) and rne(4) calls:

rn2(2) returns
rne(4) returns
Probability
Outcome
Range
Distribution

1
5

 to 

1
4

 to 

1
3

 to 

1
2

 to 

1
1

 to 

0
1

 to 

0
2

 to 

0
3

 to 

0
4

 to 

0
5

 to 

Where the ranges overlap, the distributions add:

Range
Distribution

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

This random variable, multiplied by the parameter i, is the return value of rnz.

Cumulative distribution function
The integral of the probability density function, which contains the probability that the result will be less than x:

Range
Distribution

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

Mean
If  is the probability density function of a continuous random variable, then the mean of the variable is

For the continuous random variable described by the table at the end of the previous section, we find:

Range

indefinite integral
definite integral

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

The sum of the definite integrals looks complicated at first, but by applying the laws of logarithms it can be reduced to

Thus the mean result of a call to rnz(x) is 1.29815x.  The mean prayer timeout, then, is 454.351 turns. (Actually, it is 453.787, so the error introduced by treating Z as continuous is quite small.)

Standard deviation
The standard deviation of a variable with probability density function  and mean  is:

Since  is a constant, we carry it through the calculation and substitute it at the end. The integral under the square root sign is the variance and we calculate it thus:

Range

indefinite integral
definite integral

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

 to 

Collecting terms and applying the laws of logarithms gives

and substituting the value of  gives

This is the variance of the underlying variable; the standard deviation is its square root:

The standard deviation of rnz(x), then, is 1.04424x, or 365.483 for an ordinary prayer timeout. (Actually, it is 365.344, so the error introduced by treating Z as continuous is quite small.)

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