Distributions

The Chi language has three kinds of distributions:

The constant distributions are used during creation of the Chi program. Before adding stochastic behavior, you want to make sure the program itself is correct. It is much easier to verify correctness without stochastic behavior, but if you have to change the program again after the verification, you may introduce new errors in the process.

The constant distributions solve this by allowing you to program with stochastic sampling in the code, but it is not doing anything (since you get the same predictable value on each sample operation). After verifying correctness of the program, you only need to modify the distributions that you use to get proper stochastic behavior.

Constant distributions

The constant distributions have very predictable samples, which makes them ideal for testing functioning of the program before adding stochastic behavior.

  • dist bool constant(bool b)

    Distribution always returning b.

    Range

    b
    Mean

    b
    Variance

    -

  • dist int constant(int i)

    Distribution always returning i.

    Range

    i
    Mean

    i
    Variance

    -

  • dist real constant(real r)

    Distribution always returning r.

    Range

    r
    Mean

    r
    Variance

    -

Discrete distributions

The discrete distributions return integer or boolean sample values.

  • dist bool bernoulli(real p)

    Outcome of an experiment with chance p (0 <= p <= 1).

    bernoulli
    Range

    {false, true}
    Mean

    p (where false is interpreted as 0, and true is interpreted as 1)
    Variance

    1 - p (where false is interpreted as 0, and true is interpreted as 1)
    See also

    Bernoulli(p), [Law (2007)], page 302

  • dist int binomial(int n, real p)

    Number of successes when performing n experiments (n > 0) with chance p (0 <= p <= 1).

    Range

    {0, 1, ..., n}
    Mean

    n * p
    Variance

    n * p * (1 - p)
    See also

    bin(n, p), [Law (2007)], page 304

  • dist int geometric(real p)

    Geometric distribution, number of failures before success for an experiment with chance p (0 < p <= 1).

    Range

    {0, 1, ...}
    Mean

    (1 - p) / p
    Variance

    (1 - p) / p^2
    See also

    geom(p), [Law (2007)], page 305

  • dist int poisson(real lambda)

    Poisson distribution.

    poisson
    Range

    {0, 1, ...}
    Mean

    lambda
    Variance

    lambda
    See also

    Poison(lambda), [Law (2007)], page 308

  • dist int uniform(int a, b)

    Integer uniform distribution from a to b excluding the upper bound.

    disc uni
    Range

    {a, a+1, ..., b-1}
    Mean

    (a + b - 1) / 2
    Variance

    ((b - a)^2 - 1) / 12
    See also

    DU(a, b - 1), [Law (2007)], page 303

Continuous distributions

  • dist real beta(real p, q)

    Beta distribution with shape parameters p and q, with p > 0 and q > 0.

    beta
    Range

    [0, 1]
    Mean

    p / (p + q)
    Variance

    p * q / ((p + q)^2 * (p + q + 1))
    See also

    Beta(p, q), [Law (2007)], page 291

  • dist real erlang(double m, int k)

    Erlang distribution with k a positive integer and m > 0. Equivalent to gamma(k, m / k).

    Mean

    m
    Variance

    m * m / k
    See also

    ERL(m, k), [Banks (1998)], page 153

  • dist real exponential(real m)

    (Negative) exponential distribution with mean m, with m > 0.

    exponential
    Range

    [0, infinite)
    Mean

    m
    Variance

    m * m
    See also

    expo(m), [Law (2007)], page 283

  • dist real gamma(real a, b)

    Gamma distribution, with shape parameter a > 0 and scale parameter b > 0.

    gamma
    Mean

    a * b
    Variance

    a * b^2

  • dist real lognormal(real m, v2)

    Log-normal distribution.

    lognormal
    Range

    [0, infinite)
    Mean

    exp(m + v2/2)
    Variance

    exp(2*m + v2) * (exp(v2) - 1)
    See also

    N(m, v2), [Law (2007)], page 290

  • dist real normal(real m, v2)

    Normal distribution.

    normal
    Range

    (-infinite, infinite)
    Mean

    m
    Variance

    v2
    See also

    N(m, v2), [Law (2007)], page 288

  • dist real random()

    Random number generator.

    Range

    [0, 1)
    Mean

    0.5
    Variance

    1 / 12

  • dist real triangle(real a, b, c)

    Triangle distribution, with a < b < c.

    triangle
    Range

    [a, c]
    Mean

    (a + b + c) /3
    Variance

    (a^2 + c^2 + b^2 - a*b - a*c - b*c) / 18
    See also

    Triangle(a, c, b), [Law (2007)], page 300

  • dist real uniform(real a, b)

    Real uniform distribution from a to b, excluding upper bound.

    cont uni
    Range

    [a, b)
    Mean

    (a + b) / 2
    Variance

    (b - a)^2 / 12
    See also

    U(a,b), [Law (2007)], page 282, except that distribution has an inclusive upper bound.

  • dist real weibull(real a, b)

    Weibull distribution with shape parameter a and scale parameter b, with a > 0 and b > 0.

    weibull
    Range

    [0, infinite)
    Mean

    (b / a) * G(1 / a)
    Variance

    (b^2 / a) * (2 * G(2 / a) - (1 / a) * G(1 / a)^2) with G(x) the Gamma function, G(x) = integral over t from 0 to infinity, for t^(x - 1) * exp(-t)
    See also

    Weibull(a, b), [Law (2007)], page 284

References

  • [Banks (1998)] Jerry Banks, "Handbook of Simulation: Principles, Methodology, Advances, Applications, and Practice", John Wiley & Sons, Inc., 1998, doi:10.1002/9780470172445

  • [Law (2007)] Averill M. Law, "Simulation Modeling and Analysis", fourth edition, McGraw-Hill, 2007