• probability of obtained statistic (prob);
• critical statistic for specific probability level (crit or quantile);
• random samples of distribution values (rand).

probdist  prob  F  2  12  3.46

probdist  rand  n  100

probdist  rand  z  20

> 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.

-q

Quick version. If a faster but slightly less robust/accurate version of a function exists, use that instead.

-s

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.

-L

Display limits

-O

Display options and values

-V

Display version number and date

probdist  random  z  100  |  dm  "x1*10+20"

probdist  random  u  100  |  dm  "floor(x1*10+20)"

e = epsilon (very small number), oo = infinity, p = a/b, *1--tail 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: prob|crit|rand u p|#

The uniform probability and critical value functions both return 1 minus the value.

prob u .9                     # equals 0.1
crit uni .7                   # equals 0.3
rand uniform 10               # produces 10 random numbers

BINOMIAL: prob|crit|rand b N P r|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 )

prob binomial 20 1/2 17       # is just less than 0.006
crit bin 30 3/4 .05           # equals 27 (p = .037)
rand B 40 1/4 10              # produces 10 random numbers

NORMAL-Z: prob|crit|rand n|z Z|p|#

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. The quick version of the random number generation adds twelve uniform random numbers and subtracts 6.0.

prob normal 2.0               # equals 0.977250
crit n .05                    # equals -1.644854 (one-tailed)
rand Z 10                     # produces 10 random numbers

STUDENT'S T: prob|crit|rand t df t|p|#

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.

prob t 20 2.0                 # equals 0.059
crit t-test 10 .05            # equals 2.23 (two-tailed)
rand tarantula 30 10          # produces 10 random numbers

F: prob|crit|rand f df1 df2

Probability of F-ratio computed with CACM Algorithm 322.

prob f 1 20 3.4               # equals 0.08
crit F 4 10 .05               # equals 3.48
rand F 5 30 10                # produces 10 random numbers

CHI-SQUARE: prob|crit|rand c|x df x|p|#

Probability of Chi-square computed with CACM Algorithm 299.

prob chi2 5 18                # equals 0.003
crit chi-square 2 .05         # equals 5.99
rand X 1 10                   # produces 10 random numbers