Is there a version of the Chi-Squared distribution where non-whole-number inputs are allowed? I'm curious about the transition from the PDF being convex (the yellow and green lines below) to it being concave (the rest of the lines below) between the inputs $2$ and $3$.
I was wondering for what input between these values the PDF would be a straight line. My guess is $e$, due to its ubiquitous appearances in mathematics.
If there are no fractional continuations of this distribution, are there any other ways to find out what that constant between $2$ and $3$ which produces a line would be?
An obvious answer is the Gamma distribution family $\Gamma(\alpha,\beta)$, where the two parameters are positive real numbers. (See also.) The density function for a $\Gamma(\alpha,\beta)$ random variable is $$f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}.$$ (Here $\Gamma(\alpha)$ is what it takes to make $\int_0^\infty f=1$, namely, the Gamma function.) The chi-squared distribution with $k$ degrees of freedom is a $\Gamma(\frac k 2,\frac 1 2)$ random variable, so one can understand a "chi-squared rv with $1/3$ degrees of freedom" to be a $\Gamma(1/6,1/2)$ random variable, and so on.