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In addition to the random number generators, this library provides distribution functions which map one distribution (often a uniform distribution provided by some generator) to another.
Usually, there are several possible implementations of any given mapping. Often, there is a choice between using more space, more invocations of the underlying source of random numbers, or more time-consuming arithmetic such as trigonometric functions. This interface description does not mandate any specific implementation. However, implementations which cannot reach certain values of the specified distribution or otherwise do not converge statistically to it are not acceptable.
Table 1.6. Uniform Distributions
distribution |
explanation |
example |
---|---|---|
discrete uniform distribution on a small set of integers (much smaller than the range of the underlying generator) |
drawing from an urn |
|
discrete uniform distribution on a set of integers; the underlying generator may be called several times to gather enough randomness for the output |
drawing from an urn |
|
continuous uniform distribution on the range [0,1); important basis for other distributions |
- |
|
continuous uniform distribution on some range [min, max) of real numbers |
for the range [0, 2pi): randomly dropping a stick and measuring its angle in radians (assuming the angle is uniformly distributed) |
Table 1.7. Bernoulli Distributions
distribution |
explanation |
example |
---|---|---|
Bernoulli experiment: discrete boolean valued distribution with configurable probability |
tossing a coin (p=0.5) |
|
counts outcomes of repeated Bernoulli experiments |
tossing a coin 20 times and counting how many front sides are shown |
|
measures distance between outcomes of repeated Bernoulli experiments |
throwing a die several times and counting the number of tries until a "6" appears for the first time |
|
Counts the number of failures of repeated Bernoulli experiments required to get some constant number of successes. |
flipping a coin and counting the number of heads that show up before we get 3 tails |
Table 1.8. Poisson Distributions
distribution |
explanation |
example |
---|---|---|
poisson distribution |
counting the number of alpha particles emitted by radioactive matter in a fixed period of time |
|
exponential distribution |
measuring the inter-arrival time of alpha particles emitted by radioactive matter |
|
gamma distribution |
- |
|
hyperexponential distribution |
service time of k-parallel servers each with a given service rate and probability to be chosen |
|
weibull distribution |
- |
|
extreme value distribution |
- |
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beta distribution |
- |
|
laplace distribution |
- |
Table 1.9. Normal Distributions
distribution |
explanation |
example |
---|---|---|
counts outcomes of (infinitely) repeated Bernoulli experiments |
tossing a coin 10000 times and counting how many front sides are shown |
|
lognormal distribution (sometimes used in simulations) |
measuring the job completion time of an assembly line worker |
|
chi-squared distribution |
- |
|
non-central chi-squared distribution |
- |
|
Cauchy distribution |
- |
|
Fisher F distribution |
- |
|
Student t distribution |
- |
Table 1.10. Sampling Distributions
distribution |
explanation |
example |
---|---|---|
discrete distribution with specific probabilities |
rolling an unfair die |
|
- |
- |
|
- |
- |
Table 1.11. Miscellaneous Distributions
distribution |
explanation |
example |
---|---|---|
triangle distribution |
- |
|
uniform distribution on a unit sphere of arbitrary dimension |
choosing a random point on Earth (assumed to be a sphere) where to spend the next vacations |