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PROBDIST(1) |STAT December 22, 1986

NAME
probdist - probability distribution functions, random number generation

SYNOPSIS
probdist [-v] [-s seed] [ function distribution [parameters] value ]

DESCRIPTION
_▶08◀p_▶08◀r_▶08◀o_▶08◀b_▶08◀d_▶08◀i_▶08◀s_▶08◀t is a family of probability distribution functions. _▶08◀f_▶08◀u_▶08◀n_▶08◀c_▶08◀t_▶08◀i_▶08◀o_▶08◀n_▶08◀s
include:
probability of obtained statistic (prob);
critical statistic for specific probability level (crit or quantile);
random samples of distribution values (rand).
_▶08◀d_▶08◀i_▶08◀s_▶08◀t_▶08◀r_▶08◀i_▶08◀b_▶08◀u_▶08◀t_▶08◀i_▶08◀o_▶08◀n_▶08◀s include the uniform, normal-z, binomial, chi-square F,
and t. Applicable distribution parameters, such as degrees of
freedom, should follow the name of the distribution. For example, you
can request the probability of an F-ratio:
probdist prob F 2 12 3.46
or a random sample of normal-z values:
probdist rand n 100

A single request can be supplied on the command line:
probdist rand z 20
or several can be supplied to the standard input. Blank input lines
are ignored.
> probdist
prob binomial 20 3/4 18
0.091260
critical t 8 .05
2.306004

Functions and distributions can be abbreviated with single letters;
only the first letter is used, and case does not matter. The normal-z
distribution can be requested with z or n. The chi-square
distribution can be requested with c or x.

Probabilities are between 0 and 1, usually not including those values.
When supplying probabilities to the program, they can be input in
decimal form (e.g., .05), or as a ratio of two integers (e.g., 1/20).
The ratio form must be used to specify the probability of a success in
the binomial distribution.

OPTIONS
-s seed
Supply the integer random seed for random number generation.
This value should be from 1 to the maximum integer for a system.
Otherwise, the first random seed is taken from the system clock
and process number. This option allows the replication of random
samples; the same requests with the same seed always produce the
same result.

-v Verbose output. By default, only the requested values are
printed (e.g., you ask for the probability of a t statistic and
that probability is printed, or you ask for a uniform random
sample of 10 numbers and ten numbers are printed). With the
verbose option, the output from the program contains lines with
the distribution name, parameters, value, and probability, in
order.

EXAMPLES
_▶08◀d_▶08◀m is useful for converting the output from _▶08◀p_▶08◀r_▶08◀o_▶08◀b_▶08◀d_▶08◀i_▶08◀s_▶08◀t.
Normal sample with mean 20 and standard deviation 10:
probdist random z 100 | dm x1*10+20
Uniform random integers between 20 and 29 (inclusive):
probdist random u 100 | dm floor(x1*10+20)

DISTRIBUTIONS
e = epsilon (a very small number), oo = infinity, p = a/b, *one-tailed test
params mean min max prob
Uniform 0.5 0+e 1-e x..1
Binomial N p Np 0 N x..N*
Normal Z 0 -oo +oo -oo..x*
t df see F 0 +oo x..oo
Chi-Square df df 0 +oo x..oo
F df1 df2 df2/(df2-2) 0 +oo x..oo

The critical value functions use an inversion of the probability
functions that refine their approximations until the computed
distribution value produces a probability within .000001 of the
requested value. The random samples are based on uniform random
numbers between 0.0 and 1.0 (but not including those extreme values);
the uniform random numbers are used as input to the critical statistic
calculation.

Uniform: u
The uniform probability and critical value calculations both return 1
minus the value.

Binomial: b N p
The binomial distribution returns the cumulative probability from a
given value r (number of successes) up to N (the number of trials).
The value of p, the probability of a success must be specified as a
ratio of integers (e.g., 1/2, not .5). To compute the lower tail of
the binomial distribution, that is, the probability of getting r or
less successes, the following rule can be used:
prob ( B(N,p) <= r ) = prob ( B(N,1-p) >= N-r )
For a specified significance level, such as the .05 level, there may
be no critical statistic with exactly the desired probability. In
most cases, the probability of the statistic will be less than that
requested. In some cases, there may be no critical statistic with
less than the requested probability (e.g., the probability of 5
successes in 5 binomial trials with p=1/2 is 0.03125), so the computed
value would be one greater than the maximum possible (e.g., for the
B(5,1/2) example: 6). To compute random binomial numbers, N uniform
random numbers are generated and the count of those less than p is the
random statistic; with this algorithm, under the verbose option, the
probability reported with the random statistic is not meaningful.
Probability calculations are based on a logarithmic approximation of
sums of products of powers of primes, thought to be accurate to over
ten decimal places for N up to 1000.

Normal-Z: n
The normal-z probability function computes values for the one-tailed
cumulative probability from -oo up to the given value. The function
is accurate to six decimal places (z values with absolute values up to
6). Probability of Normal Z value computed with CACM Algorithm 209.

Student's t: t df
The probability of Student's t-value computed from the relation:
t(n)*t(n) = F(1,n) so, the probability reported for a t statistic is
the two-tailed probability of |t| exceeding the obtained value.

F: f df1 df2
Probability of F-ratio computed with CACM Algorithm 322.

Chi-Square: c df
Probability of Chi-square computed with CACM Algorithm 299.

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Derivation

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