Cumulative distribution function of the chi-square distribution
This functionality does not run in MATLAB.
stats::chisquareCDF(m) returns a procedure
representing the cumulative distribution function
of the chi-square distribution with mean m > 0.
f := stats::chisquareCDF(m) can
be called in the form
f(x) with an arithmetical
x. The return value of
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 m can be converted to a positive floating-point number, then these values are returned as floating-point numbers. Otherwise, symbolic expressions are returned.
f reacts to properties of identifiers
If x is
a symbolic expression with the property x ≤
0 or x ≥
0, the corresponding values are returned.
f(x) returns the symbolic call
neither x ≤ 0 nor x >
0 can be decided.
Numerical values for
m are only accepted
if they are real and positive.
Note that, for large
m, exact results may
be costly to compute. If floating-point values are desired, it is
recommended to pass floating-point arguments
than to compute exact results
f(x) and convert
Cf. Example 4.
The function is sensitive to the environment variable
determines the numerical working precision.
We evaluate the cumulative distribution function with mean m = 2 at various points:
f := stats::chisquareCDF(2): f(-infinity), f(-3), f(1/2), f(0.5), f(PI), f(infinity)
x is a symbolic object without properties,
then it cannot be decided whether x ≥
0 holds. A symbolic function call is returned:
f := stats::chisquareCDF(m): f(x)
With suitable properties, it can be decided whether x ≥ 0 holds. An explicit expression is returned:
assume(0 <= x): f(x)
For integer values of
m, the special function
igamma can be expressed
in terms of more elementary functions:
m := 6: f(x)
m := 5: f(x)
unassume(x): delete f, m:
We use a symbolic mean
f := stats::chisquareCDF(m): f(3), f(3.0)
When a numerical value is assigned to
f starts to produce numerical values:
m := PI: f(3), f(3.0)
delete f, m:
We consider a chi-square distribution with large mean
m = 1000:
f := stats::chisquareCDF(1000):
For floating-point approximations, one should not compute an
exact result and convert it via
float. For large mean m,
it is faster to pass a floating-point argument to
The following call takes some time, because an exact computation of
the huge integer
gamma(m/2) = gamma(500) = 499! is
The following call is much faster:
The mean: an arithmetical expression representing a positive real value