| DiscreteGeneralDistribution |
General Discrete Distribution
| Meaning:
The first \(n\) values \(p_i (i = 0, ...
|
| DiscreteUniformDistribution |
Discrete Uniform Distribution
| Meaning:
All integer values \(i\) in the interval \(b_l \le i < b_u \) (\(b_l\)
and \(b_u\) being also integer values) have the same probability
\(\frac{1}{b_u-b_l}\)
| Parameters:
lower limit \(b_l\)
upper limit \(b_u > b_l\)
Distribution:
\(P(X=i) = \begin{cases} \frac{1}{b_u-b_l} &\mbox{for } b_l \le i < b_u
\\ 0 & \mbox{else } \end{cases} \)
| Expected value:
\(E[X]=\frac{b_l+b_u-1}{2} \)
| Variance:
\(VAR[X]=\frac{(b_u-b_l-1) \cdot (b_u-b_l+1)}{12}\)
| Coefficient of variation:
\(c_X=\frac{ \sqrt{ (b_u-b_l-1) \cdot (b_u-b_l+1) } }{ \sqrt{3} \cdot
(b_l+b_u-1)}\)
| Generating func.:
\(G(z)=z^{b_l}+...+z^{b_u-1} \)
| Parser example:
[...].Distribution = DiscreteUniform
[...].Distribution.LowerBound = 3
[...].Distribution.UpperBound = 7
|
| GeometricDistribution |
Geometric Distribution
| Meaning:
Probability for \(i\) failures prior to the first success in independent
Bernoulli experiments with the parameter \(q(0 \le q \le 1)\)
| Parameters:
success probability \(q(0 < q \le 1)\)
With mean parameter \(m : q = \frac{1}{1+m}\)
Distribution:
\(P(X=i) = (1-q)^i \cdot q = (\frac{m}{m+1})^i \cdot \frac{1}{m+1}
\mbox{for } i = 0,1,2,...
|
| NegBinDistribution |
Negative Binomial Distribution
| Meaning:
Probability, that \(i\) failed Bernoulli trials with the parameter \(q\)
will precede \(k\) successes
| Parameters:
success probability \(q(0 \le q \le 1)\)
number of successful trial \(k \ge 0\)
With mean value and variance parameters:
\(q= \frac{VAR[X]}{E[X]} \)
\(k= \frac{E[X]}{VAR[X] - E[X]}\)
Distribution:
\(P(X=i) = \binom{i+k-1}{k-1} \cdot q^k \cdot (1-q)^i = \binom{i+
\frac{m^2}{v-m} -1}{ \frac{m^2}{v-m} -1} \cdot
(\frac{m}{v})^{\frac{m^2}{v-m}} \cdot (1- \frac{m}{v})^i \)
| Expected value:
\(E[X]= \frac{1-q}{q} \cdot k = m\)
| Variance:
\(VAR[X]= \frac{1-q}{q^2} \cdot k = \frac{m}{q} = v\)
| Coefficient of variation:
\(c_T= \frac{1}{ \sqrt{(1-q) \cdot k}}\)
| Generating func.:
\(G(z)= \frac{q^k}{(1-(1-q) \cdot z)^k} \)
| Parser example:
[...].Distribution = NegBin
[...].Distribution.Mean = 5
[...].Distribution.Variance = 10
|
| PoissonDistribution |
Poisson Distribution
| Meaning:
Probability of the number of arrivals in a time interval with the
duration \(t\) for a Markovian arrival process (limit distribution of a
binomial distribution for \(n \to \infty , q \to 0, nq \to \lambda t\))
| Parameters:
mean value \(m = \lambda t > 0\)
Distribution:
\(P(X=i) = \frac{(\lambda t)^i}{i!} \cdot exp(-\lambda t) =
\frac{m^i}{i!} \cdot exp(-m) \)
| Expected value:
\(E[X]= \lambda t = m \)
| Variance:
\(VAR[X]= \lambda t = m\)
| Coefficient of variation:
\(c_T=\frac{1}{ \sqrt{ \lambda t }} = \frac{1}{ \sqrt{m}}\)
| Generating func.:
\(G(z)= exp(-\lambda t \cdot (1-z)) = exp(-m \cdot (1-z))\)
| Parser example:
[...].Distribution = Poisson
[...].Distribution.Mean = 2.5
|
| ShiftedGeometricDistribution |
Shifted Geometric Distribution
| Meaning:
Probability for \(i-1\) failures prior to the first success in
independent Bernoulli experiments with the parameter \(q(0 \le q \le 1)\)
| Parameters:
Success probability \(q\)
With mean value parameter \(m : q = \frac{1}{m}\)
Distribution:
\(P(X=i) = (1-q)^{i-1} \cdot q = (\frac{m-1}{m})^i \cdot \frac{1}{m-1}
\mbox{for } i = 1,2,...
|
| ZipfDistribution |
Zipf Distribution ("Discrete Pareto Distribution")
|
| ZipfMandelbrotDistribution |
Zipf-Mandelbrot Distribution
|
Package ikr.simlib.distributions.discrete Description
Discrete Distributions
Classes, that have been derived from DiscreteDistribution, override the abstract
method next(), which returns an integer value. | | | | | |