KermMartian at 19:24, 24 February 2016

2016-02-24T19:24:51Z

← Older revision		Revision as of 19:24, 24 February 2016
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	Alternatively, you can exploit the identity		Alternatively, you can exploit the identity

	~~[[TI-BASIC:Math\|Math]]~~		<math>
	\operatorname{tcdf}(-\infty,0,\nu)=\frac1{2}		\operatorname{tcdf}(-\infty,0,\nu)=\frac1{2}
	~~[[TI-BASIC:~~/math~~\|/math]]~~		</math>

	(similarly for the interval from 0 to ∞)		(similarly for the interval from 0 to ∞)
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	and thus		and thus

	~~[[TI-BASIC:Math\|Math]]~~		<math>
	\operatorname{tcdf}(-\infty,x,\nu)=\frac1{2}+\operatorname{tcdf}(0,x,\nu)		\operatorname{tcdf}(-\infty,x,\nu)=\frac1{2}+\operatorname{tcdf}(0,x,\nu)
	~~[[TI-BASIC:~~/math~~\|/math]]~~		</math>

	For the form used in two-tailed tests, the following identity may be useful:		For the form used in two-tailed tests, the following identity may be useful:

	~~[[TI-BASIC:Math\|Math]]~~		<math>
	\operatorname{tcdf}(-x,x,\nu)=2\operatorname{tcdf}(0,x,\nu)		\operatorname{tcdf}(-x,x,\nu)=2\operatorname{tcdf}(0,x,\nu)
	~~[[TI-BASIC:~~/math~~\|/math]]~~		</math>

	= Formulas =		= Formulas =
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	As with any other continuous distribution, tcdf( can be defined in terms of the probability density function, [[TI-BASIC:Tpdf\|Tpdf(]]:		As with any other continuous distribution, tcdf( can be defined in terms of the probability density function, [[TI-BASIC:Tpdf\|Tpdf(]]:

	~~[[TI-BASIC:Math\|Math]]~~		<math>
	\operatorname{tcdf}(a,b,\nu)=\int_a^b \operatorname{tpdf}(t,\nu)\mathrm{d}t		\operatorname{tcdf}(a,b,\nu)=\int_a^b \operatorname{tpdf}(t,\nu)\mathrm{d}t
	~~[[TI-BASIC:~~/math~~\|/math]]~~		</math>

	The function can also be expressed in terms of an [http://mathworld.wolfram.com/IncompleteBetaFunction.html incomplete beta function].		The function can also be expressed in terms of an [http://mathworld.wolfram.com/IncompleteBetaFunction.html incomplete beta function].
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	For one degree of freedom (ν=1), tcdf( is expressible in terms of simpler functions:		For one degree of freedom (ν=1), tcdf( is expressible in terms of simpler functions:

	~~[[TI-BASIC:Math\|Math]]~~		<math>
	\operatorname{tcdf}(a,b,1)=\frac1{\pi}\left(\tan^{-1}\left(b\right)-\tan^{-1}\left(a\right)\right)		\operatorname{tcdf}(a,b,1)=\frac1{\pi}\left(\tan^{-1}\left(b\right)-\tan^{-1}\left(a\right)\right)
	~~[[TI-BASIC:~~/math~~\|/math]]~~		</math>

	This is the so-called Cauchy distribution.		This is the so-called Cauchy distribution.

Maintenance script: Initial automated import

2016-02-24T18:36:05Z

Initial automated import

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{{Template:TI-BASIC:Command
|picture=TCDF.GIF
|summary=Calculates the Student's ''t'' probability betwen ''lower'' and ''upper'' for degrees of freedom ν.
|syntax=tcdf(''lower'', ''upper'', ''ν'')
|location=Press:
# 2ND DISTR to access the distribution menu
# 5 to select tcdf(, or use arrows.
Press 6 instead of 5 on a TI-84+/SE with OS 2.30 or higher.
|compatibility=TI-83/84/+/SE
|size=2 bytes
}}

tcdf( is the Student's ''t'' cumulative density function. If some random variable follows this distribution, you can use this command to find the probability that this variable will fall in the interval you supply.

Unlike [[TI-BASIC:Normalcdf|Normalcdf(]], this command only works for the standardized distribution with mean 0 and standard deviation 1. To use it for non-standardized values you will have to standardize them by calculating (X-μ)/σ (where μ is the mean and σ the standard deviation). Do this for both ''lower'' and ''upper''.

= Advanced =

Often, you want to find a "tail probability" - a special case for which the interval has no lower or no upper bound (the form frequently used in one-tailed tests). For example, "what is the probability x is greater than 2?". The TI-83+ has no special symbol for infinity, but you can use E99 to get a very large number that will work equally well in this case ([[TI-BASIC:E_Ten|E]] is the decimal exponent obtained by pressing [2nd] [EE]). Use E99 for +∞, and -E99 for -∞.

Alternatively, you can exploit the identity

[[TI-BASIC:Math|Math]]
\operatorname{tcdf}(-\infty,0,\nu)=\frac1{2}
[[TI-BASIC:/math|/math]]

(similarly for the interval from 0 to ∞)

and thus

[[TI-BASIC:Math|Math]]
\operatorname{tcdf}(-\infty,x,\nu)=\frac1{2}+\operatorname{tcdf}(0,x,\nu)
[[TI-BASIC:/math|/math]]

For the form used in two-tailed tests, the following identity may be useful:

[[TI-BASIC:Math|Math]]
\operatorname{tcdf}(-x,x,\nu)=2\operatorname{tcdf}(0,x,\nu)
[[TI-BASIC:/math|/math]]

= Formulas =

As with any other continuous distribution, tcdf( can be defined in terms of the probability density function, [[TI-BASIC:Tpdf|Tpdf(]]:

[[TI-BASIC:Math|Math]]
\operatorname{tcdf}(a,b,\nu)=\int_a^b \operatorname{tpdf}(t,\nu)\mathrm{d}t
[[TI-BASIC:/math|/math]]

The function can also be expressed in terms of an [http://mathworld.wolfram.com/IncompleteBetaFunction.html incomplete beta function].

For one degree of freedom (ν=1), tcdf( is expressible in terms of simpler functions:

[[TI-BASIC:Math|Math]]
\operatorname{tcdf}(a,b,1)=\frac1{\pi}\left(\tan^{-1}\left(b\right)-\tan^{-1}\left(a\right)\right)
[[TI-BASIC:/math|/math]]

This is the so-called Cauchy distribution.

= Related Commands =

* [[TI-BASIC:Tpdf|Tpdf(]]
* [[TI-BASIC:Invt|InvT(]]
* [[TI-BASIC:Shade_T|Shade_T(]][[Category:TI-BASIC]]
[[Category:TIBD]]

TI-BASIC:Tcdf - Revision history

KermMartian at 19:24, 24 February 2016

Maintenance script: Initial automated import