The (discrete) cumulative distribution function of the binomial distribution
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stats::binomialCDF(n
, p
)
stats::binomialCDF(n, p)
returns a procedure
representing the (discrete) cumulative distribution function
.
of the binomial distribution with "trial parameter" n
and
"probability parameter" p
.
The procedure f := stats::binomialCDF(n, p)
can
be called in the form f(x)
with an arithmetical
expression x
. The return value of f(x)
is
either a floatingpoint number, an exact numerical value, or a symbolic
expression:
If x
is a numerical real value
and n
is a positive integer, then an explicit value
is returned. If p
is a numerical value satisfying 0
≤ p ∧ p ≤
1, this is a numerical value. Otherwise, it is
a symbolic expression in p
.
If x
is a numerical value with x <
0, then 0
, respectively 0.0
,
is returned for any value of n
and p
.
For symbolic values of n
, explicit
results are returned if x is
a numerical value with x <
2.
For symbolic values of n
, explicit
results are returned if n  x is
a numerical value with n  x ≤
2.
If n  x
is a numerical value with n  x ≤
0, then 1
, respectively 1.0
,
is returned for any value of n
and p
.
In all other cases, f(x)
returns
the symbolic call binomialCDF(n, p)(x)
.
Numerical values for n
are only accepted
if they are positive integers.
Numerical values for p
are only accepted
if they satisfy 0 ≤ p ≤
1.
If x
is a real floatingpoint number, the
result is a floating number provided n and p are
numerical values. If x
is an exact numerical value,
the result is an exact number.
Note:
Note that for large n,
floatingpoint results are computed much faster than exact results.
If floatingpoint approximations are desired, pass a floatingpoint
number 
The function is sensitive to the environment variable DIGITS
which
determines the numerical working precision.
We evaluate the distribution function with n = 20 and at various points:
f := stats::binomialCDF(5, 3/4): f(1), f(2), f(PI), f(5), f(6)
f(1.2), f(2.0), f(float(PI)), f(5.5)
delete f:
We use symbolic arguments:
f := stats::binomialCDF(n, p): f(x), f(8), f(8.0)
When numerical values are assigned to n and p, the function f starts to produce explicit results if the argument is numerical:
n := 3: p := 1/3: f(2), f(2.5), f(PI +1), f(4.0)
delete f, n, p:
If n and x are numerical, symbolic expressions are returned for symbolic values of p:
f := stats::binomialCDF(3, p): f(1), f(0), f(3/2), f(1 + sqrt(3)), f(2.999), f(3)
delete f:

The "trial parameter": an arithmetical expression representing a positive integer 

The "probability parameter": an arithmetical expression representing a real number 0 ≤ p ≤ 1. 