Probability function of the binomial distribution
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stats::binomialPF(n
, p
)
stats::binomialPF(n, p)
returns a procedure
representing the probability function
for x=0,1,..,n
of the binomial distribution
with "trial parameter" n
and "probability
parameter" p
.
The procedure f := stats::binomialPF(n, p)
can
be called in the form f(x)
with arithmetical expressions x
.
The return value of f(x)
is either a floatingpoint
number, an exact numerical value, or a symbolic expression:
If x
is a noninteger numerical
value, f(x) returns 0
or 0.0
,
respectively.
If x
is an integer or the floating
point equivalent of an integer and n
is a positive
integer, then an explicit value is returned. If p
is
a numerical value satisfying 0 ≤ p ≤
1, this is a numerical value. Otherwise, it is
a symbolic expression in p
.
For symbolic values of n
, explicit results
are returned if x is
a numerical value withx <
2.
For symbolic values of n
, explicit
results are returned if n  x is
a numerical value withn  x <
2.
In all other cases, f(x)
returns
the symbolic call stats::binomialPF(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 floatingpoint number, the result
is a floating number provided n and p are
numerical values. If x
is an exact value, the result
is an exact number.
Note that for large n,
floatingpoint results are computed much faster than exact results.
If floatingpoint approximations are desired, pass a floatingpoint
number x
to the procedure created by stats::binomialPF
.
The function is sensitive to the environment variable DIGITS
which
determines the numerical working precision.
We compute the probability function with n = 3 and at various points:
f := stats::binomialPF(3, 3/4): f(1/2), f(0), f(1/2), f(1), f(7/4), f(2), f(3), f(4)
f(0.2), f(0.0), f(0.7), f(1.0), f(2.0), f(2.7), f(3.0), f(4.0)
delete f:
We use symbolic arguments:
f := stats::binomialPF(n, p): f(x), f(8), f(8.0)
When real numbers 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(0), f(1), f(2.0), f(3.5), f(4)
delete f, n, p, x:
If n and x are numerical, symbolic expressions are returned for symbolic values of p:
f := stats::binomialPF(3, p): f(1), f(0), f(3/2), f(2), 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. 