nonCentralTPdf¶
Evaluates the noncentral Student’s t probability density function.
Synopsis¶
nonCentralTPdf (t, df, delta)
Required Arguments¶
- float
t
(Input) - Argument for which the noncentral Student’s t probability density function is to be evaluated.
- float
df
(Input) - Number of degrees of freedom of the noncentral Student’s t
distribution.
df
must be greater than 0. - float
delta
(Input) - Noncentrality parameter.
Return Value¶
The probability density associated with a noncentral Student’s t random
variable with value t
.
Description¶
If w is a normally distributed random variable with unit variance and mean δ and u is a chi-square random variable with ν degrees of freedom that is statistically independent of w, then
is a noncentral t-distributed random variable with ν degrees of freedom
and noncentrality parameter δ, that is,with ν = df
, and δ = delta
.
The probability density function for the noncentral t-distribution is:
where
and t = t
.
For \(\delta=0\), the PDF reduces to the (central) Student’s t PDF:
and, for \(t=0\), the PDF becomes:
Example¶
This example calculates the noncentral Student’s t PDF for a distribution with 2 degrees of freedom and noncentrality parameter \(\delta=10\).
from __future__ import print_function
from pyimsl.stat.nonCentralTPdf import nonCentralTPdf
t = [-.5, 1.5, 3.5, 7.5, 51.5, 99.5]
df = 2.
delta = 10.
print("\n df: %4.0f; delta: %4.0f\n" % (df, delta))
print(" t pdf(t)")
for i in range(0, 6):
pdfv = nonCentralTPdf(t[i], df, delta)
print(" %5.1f %12.4e" % (t[i], pdfv))
Output¶
df: 2; delta: 10
t pdf(t)
-0.5 1.6399e-24
1.5 7.4417e-10
3.5 2.8972e-03
7.5 7.8853e-02
51.5 1.4215e-03
99.5 2.0290e-04