- #1

Dustinsfl

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With a normal distribution, I know the equation is y=e^(-x^2)/sqr(2*pi).

Is there a t distribution I can integrate for a t distribution?

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- Thread starter Dustinsfl
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- #1

Dustinsfl

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With a normal distribution, I know the equation is y=e^(-x^2)/sqr(2*pi).

Is there a t distribution I can integrate for a t distribution?

- #2

coreyB

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- #3

statdad

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I'm not sure what you're after - if you use software the p-value will be (is, for the software with which I'm aware) reported with the output. if you are working by hand, you should know how to use tables to find, or at least approximate, p-values.

- #4

Dustinsfl

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- #5

statdad

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There is no closed form integral for the t-distribution density (of course there isn't for the normal distribution's density either). (Actually, the cumulative function involves a hypergeometric function).

The density for the t-distribution that has [itex] k [/itex] degrees of freedom is

[tex]

f(x) = \frac{\Gamma\left(\frac{k+1}{2}\right)}{\sqrt{\, \k \pi} \Gamma\left(\frac k 2\right)}\left(1 + \frac {x^2} k\right)^{-\left(\frac{k+1}2\right)}

[/tex]

The density for the t-distribution that has [itex] k [/itex] degrees of freedom is

[tex]

f(x) = \frac{\Gamma\left(\frac{k+1}{2}\right)}{\sqrt{\, \k \pi} \Gamma\left(\frac k 2\right)}\left(1 + \frac {x^2} k\right)^{-\left(\frac{k+1}2\right)}

[/tex]

Last edited:

- #6

Dustinsfl

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What is gamma?

- #7

statdad

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[itex] \Gamma [/itex] is the gamma function.

- #8

Dustinsfl

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Is the gamma function defined by a formula with unknowns? And if so, what is it?

- #9

statdad

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[tex]

\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \, dt

[/tex]

The integral converges for [itex] x > 1 [/itex]. (It can be defined for complex values as well,

but that isn't needed for your question.)

Is the gamma function defined by a formula with unknowns? And if so, what is it?

If [itex] x [/itex] is a positive integer it can be shown that

[tex]

\Gamma(x) = (x-1)!

[/tex]

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