Please help, I am way above my head in the math at this point. I have had some calculus and some statistics. I am more than willing to learn, but I am having a huge confusion right now.
In "Mathematical Statistics with Applications" by Wackerly et. al. they state:
"If $\alpha$ is not an integer and 0 < c < d < $\infty$, it is impossible to give a closed-form expression for:
$$ \int_c^d {{y}^{\alpha-1}e^{-y\over \beta}\over \beta^\alpha \Gamma(\alpha)}dy $$
As a result, it is impossible to obtain areas under the gamma density function by direct integration. Tabulated values for integrals like the above are given in Tables of the Incomplete Gamma Function (Pearson 1965)."
I have an integral that's similar to this, except messier, and I'm trying to do some basic research on how to solve it. From other answers I found this Wikipedia article "List of integrals of exponential functions" https://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions#Integrals_involving_polynomials where it appears that the 3rd formula does show a closed-form solution for the gamma integral, if you let n = $\alpha$ - 1 and c = -1/$\beta$. They say that the solution is:
$$ \int x^ne^{cx}~=~e^{cx}\sum_{i=0}^n (-1)^{n-i}{n!\over i!c^{n-i+1}x^i} $$
But (while typing my question lol) I see n must be an integer here, and $\alpha$ is not an integer in general... ($\alpha$ = v/2, where v = degrees of freedom)
- So if alpha is not an integer then how do we evaluate this integral? I guess in my case I will only need it for multiples of 1/2.
- Does anyone know a good resource (Calculus textbook) that goes over these type of integrals in detail? How they are derived etc.? Does it have something to do with differential equations, because I noticed a partial derivative in one of the steps from the Wiki article.
Any help is much appreciated!
Note that the denominator is a constant and can be factored out of the integral.
Mathematica (actual software) gives:
$$\frac{\left(\frac{1}{\beta}\right)^{-\alpha} \beta^{-\alpha} \left(\Gamma \left(\alpha,\frac{c}{b}\right)-\Gamma \left(\alpha,\frac{d}{\beta}\right)\right)}{\Gamma (\alpha)}$$
which checks out numerically for several choices of $\alpha, \beta, c,$ and $d$.