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Cumulative distribution function of the Erlang distribution

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stats::erlangCDF(a, b)


stats::erlangCDF(a, b) returns a procedure representing the cumulative distribution function

of the Erlang distribution with shape parameter a > 0 and scale parameter b > 0.

The procedure f := stats::erlangCDF(a, b) can be called in the form f(x) with an arithmetical expression x. The return value of f(x) is either a floating-point number or a symbolic expression:

If x ≤ 0 can be decided, then f(x) returns 0. If x ≥ 0 can be decided, then f(x) returns the value .

If x is a floating-point number and both a and b can be converted to positive floating-point numbers, then these values are returned as floating-point numbers. Otherwise, symbolic expressions are returned.

The function f reacts to properties of identifiers set via assume. If x is a symbolic expression with the property x ≤ 0 or x ≥ 0, the corresponding values are returned.

The call f(- infinity ) returns 0.

The call f( infinity ) returns 1.

f(x) returns the symbolic call stats::erlangCDF(a, b)(x) if neither x ≤ 0 nor x ≥ 0 can be decided.

Numerical values for a and b are only accepted if they are real and positive.

Note that, for large a, exact results may be costly to compute. If floating-point values are desired, it is recommended to pass floating-point arguments x to f rather than to compute exact results f(x) and convert them via float. Cf. Example 4.

Note that .

Environment Interactions

The function is sensitive to the environment variable DIGITS which determines the numerical working precision. The procedure generated by stats::erlangCDF reacts to properties of identifiers set via assume.


Example 1

We evaluate the cumulative probability function with a = 2 and b = 1 at various points:

f := stats::erlangCDF(2, 1):
f(-infinity), f(-3), f(0.5), f(2/3), f(PI), f(infinity)

delete f:

Example 2

If x is a symbolic object without properties, then it cannot be decided whether x ≥ 0 holds. A symbolic function call is returned:

f := stats::erlangCDF(a, b): f(x)

With suitable properties, it can be decided whether x ≥ 0 holds. An explicit expression is returned:

assume(0 <= x): f(x)

unassume(x): delete f:

Example 3

We use symbolic arguments:

f := stats::erlangCDF(a, b): f(3), f(3.0)

When numerical values are assigned to a and b, the function f starts to produce numerical values:

a := 2: b := 4: f(3), f(3.0)

delete f, a, b:

Example 4

We consider an Erlang distribution with large shape parameter:

f := stats::erlangCDF(2000, 2):

For floating-point approximations, one should not compute an exact result and convert it via float. For large shape parameter, it is faster to pass a floating-point argument to f. The following call takes some time, because an exact computation of the huge integer is involved:


The following call is much faster:


delete f:



The shape parameter: an arithmetical expression representing a positive real value


The scale parameter: an arithmetical expression representing a positive real value

Return Values


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