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 floatingpoint 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 floatingpoint number and both a and b can be converted to positive floatingpoint numbers, then these values are returned as floatingpoint 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 floatingpoint values are
desired, it is recommended to pass floatingpoint arguments x
to f
rather
than to compute exact results f(x)
and convert
them via float
.
Cf. Example 4.
Note that .
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
.
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:
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:
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:
We consider an Erlang distribution with large shape parameter:
f := stats::erlangCDF(2000, 2):
For floatingpoint approximations, one should not compute an
exact result and convert it via float
. For large shape parameter, it
is faster to pass a floatingpoint argument to f
.
The following call takes some time, because an exact computation of
the huge integer
is
involved:
float(f(1010))
The following call is much faster:
f(float(1010))
delete f:

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

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