Cdf

It is a cumulative distribution function (cdf):

F_Z :     z $$\longmapsto$$ Prob{Z$$\le$$z}

It can be either defined analytically (e.g. a Gaussian cdf, or an exponential cdf) or by a family of couples $\Big($z_i, F_Z(z_n)$\Big)$, i = 1,..., n (non-parametric cdf).

Associated Types

• Value Type

  A::value_type

  The type of z (see definition above).

Notations

A		A type that is a model of CDF
a		Object of type X
z		Object of type X::value_type
P		A type that represents a real number (e.g. float, double)
p		an object of type P

Valid Expressions

• Cdf Evaluation

  a.prob(z)

  Return type:		P
  Semantics:		returns p = Prob{Z$\le$z}
  Postcondition:		0$\le$p$\le$1 (p is a probability)

• Cdf Inverse

  a.inverse(p)

  Precondition:		0$\le$p$\le$1 (p is a probability)
  Return type:		X::value_type
  Semantics:		returns the value z such that p = Prob{Z$\le$z}

Definition

A Cdf is valid if

•     $\forall$z    F_Z(z) $\in$ [0, 1].
• F_Z is a monotonous, increasing function.

This means for example that order relation problems (resulting from some indicator-based algorithms) must be corrected before the object is actually a valid cdf.

Models

• gaussian_cdf
• discrete_variable_non_parametric_cdf