# Noncentral t-distribution Calculator

## Calculates the probability density function and lower and upper cumulative distribution functions of the noncentral t-distribution.

 percentile x degree of freedom ν ν＞0 noncentrality λ λ≧0 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit
 ・when λ=0, this comes to the student's t-distribution.$\normal Noncentral\ Student's\ t{\tiny-}distribution\\\hspace{240}t(x,\nu,\lambda)\\[10](1)\ probability\ density\\\ f(x,\nu,\lambda)={\small-}{\large\sum_{\small j=0}^{\small \infty}\frac{e^{-\frac{\lambda}{2}}(\frac{\lambda}{2})^j}{j!}\frac{(\frac{\nu}{\nu+x^2})^{\frac{\nu}{2}}(\frac{x^2}{\nu+x^2})^{\frac{1}{2}+j}}{B(\frac{\nu}{2},\frac{1}{2}+j)}\frac{2}{x}}\\\hspace{75}+{\large\sum_{\small j=\frac{1}{2}}^{\small \infty}\frac{e^{-\frac{\lambda}{2}}(\frac{\lambda}{2})^j}{j!}\frac{(\frac{\nu}{\nu+x^2})^{\frac{\nu}{2}}(\frac{x^2}{\nu+x^2})^{\frac{1}{2}+j}}{B(\frac{\nu}{2},\frac{1}{2}+j)}\frac{2}{x}}\\(2)\ lower\ cumulative\ distribution\\\hspace{25}P(x,\nu,\lambda)={\large\int_{\small-\infty}^{\hspace{25}\small x}}f(t,\nu,\lambda)dt\\(3)\ upper\ cumulative\ distribution\\\hspace{25}Q(x,\nu,\lambda)={\large\int_{\small x}^{\hspace{25pt}\small\infty}}f(t,\nu,\lambda)dt\\$

Noncentral t-distribution
 [1-7] /7 Disp-Num5103050100200
[1]  2018/11/26 04:14   Male / 60 years old level or over / A teacher / A researcher / Not at All /
Purpose of use
Certified values for evaluating algorithms
Comment/Request
It would be very useful if reliable; but not very, since it is currently unable to recognize when it is in trouble.

A year and a half ago, another user reported an example where the reported pdf was negative, which is of course impossible. Not only has that not been fixed in the last year and a half, the reported cdf from that example is easily recognized as ridiculous. It is well know that when t<0 and ncp>0, then F(t,df,ncp) < P(-ncp), where F is the noncentral t cdf, and P is the standard normal cdf. For that posted example (t=-65.85334569, df=9, ncp=31.6227766), that means that F(t,df,ncp) < 9e-220, yet you report it to be 0.0044...

Your calculator does work with t*ncp<0 (e.g. t=-1, df=1, ncp=1), but without more information it is difficult to know when it can be trusted.
[2]  2017/05/12 03:04   Male / 60 years old level or over / Others / Useful /
Purpose of use
Wanted to use it as correct results to compare accuracy of other algorithms, but it appears to be unreliable.
Comment/Request
Incorrect result
t = -65.85334569
df = 9
ncp = 31.6227766
with 22 digits, reports pdf < 0
[3]  2017/02/28 00:35   Female / 20 years old level / High-school/ University/ Grad student / Useful /
Purpose of use
To find critical T values and try and calculate power
Comment/Request
Please provide power for the input values.
from Keisan
You can use power by inputting ex) [3^2].
[4]  2017/02/11 12:52   Male / 60 years old level or over / High-school/ University/ Grad student / Very /
Purpose of use
Compare with own method.
[5]  2010/05/06 02:48   - / - / A researcher / Very /
Purpose of use
power analysis
Comment/Request
A wonderful public service. Thanks!
[6]  2010/03/24 22:25   Male / More than 60 / Self-employed people / Very /
Purpose of use
Work related calculation.
Comment/Request
Thanks
[7]  2010/01/16 09:09   Male / More than 60 / A researcher / Very /
Comment/Request
would like to be able to copy formula to a paper I am writing

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