With a t distribution, can I find the p value?

With a t distribution, can I find the p value?

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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yes, you can find the p-value of a t distribution. the p-value is the probability of getting something as extreme or more extreme than your test statistic. So, ff you know your test statistic, you can find the p-value.

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If you are dealing with tests for one mean, each different sample size has a different t-distribution.
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.

I want to be able to form the equation of the distribution and then integrate to solve for the p value.

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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 $k$ degrees of freedom is

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

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What is gamma?

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$\Gamma$ is the gamma function.

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

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The general definition is

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

The integral converges for $x > 1$. (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 $x$ is a positive integer it can be shown that

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