Class Binomial

_boundaries

inverseCumulativeDistribution

• inverseCumulativeDistribution(n): number

• Generates an observation using the inverse cumulative distribution method.

  This method uses:

  • the cumulative distribution function to breakdown its from/input set into observations zone
  • a uniform random function to generate a number x, that will be used to identify which observation zone is chosen.

  The process is the following:

  1- The cumulative distribution function is used to define boundaries of observation zone. 2- An number x is drawn using a uniform distribution function 3- The zone in which the number x falls is the observation

  This process can be describe using the following schema with:

  • observations in ordinate (from 0 to 4)
  • probability for each observation in abscissa

  0 ─ 1 ── 2 ──── 3 ─ 4 ───

  Probabilities are cumulated:

  0 ─ 1 ── 2 ──── 3 ─ 4 ───

  Cumulated probabilities are projected on the same dimension:

  ├┼─┼───┼┼──┤ 0 1 2 3 4

  x is drawn and the corresponding observation zone is identified using boundaries:

  ├┼x┼───┼┼──┤ 0 1 2 3 4

  Parameters

  • n: number

    Sampling upper limit.

  Returns number

  Observation

rejectionSampling

• rejectionSampling(n, max): number

• Generates an observation using the rejection sampling method.

  This method uses a uniform random function to generate:

  • the number k, the potential observation
  • the number x, a random number between 0 and the greatest probability of the distribution

  If the number x is lower or equal to the probability of event k, then it's an observation.

  Otherwise, the process is restarted (k and x are generated).

  This process can be describe using the following schema with:

  • observations in abscissa (from 0 to 4)
  • probability for each observation in ordinate (observation 2 having the greatest probability p_max)

  p_max ────┬─┬──── │ │ ┌─┐ ┌─┤ │ │ │ ┌─┤ │ ├─┤ │ └─┴─┴─┴─┴─┘ 0 1 2 3 4

  Rejection sampling method for this example do the following:

  1- randomly find the potential observation k by using an uniform random function with lower limit 0 and upper limit 4. Lets say k = 0 2- draw a number x, from an uniform random function with lower limit 0 and upper limit p_max. Lets say x > p_0. In that case we restart the process from the beginning. 3- randomly find k. Lets say k = 2 4- randomly find x. Lets say x < p_2. In that case the process stop and the observation is returned.

  Graphically, the following process can be represented as the following:

  p_max ────┬─┬──── │o│ ┌─┐ x┌─┤ │ │ │ ┌─┤ │ ├─┤ │ └─┴─┴─┴─┴─┘ 0 1 2 3 4

  Where x represents the failed attempt and o the success one.

  Intuitively we "see" the returned observations will match the underlying probabilities because observations with greater probability will have a higher chance of being returned than observation with lower one.

  Parameters

  • n: number

    sampling upper limit

  • max: number

    greatest probability of the distribution

  Returns number

  an observation