Interpretation for changes in a $\chi^2$'s density as $k$ increases The chi-square's density becomes more regular as $k$ increases:


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*$k=1$ unbounded, convex

*$k=2$ bounded, convex

*$k=3$ close to 0 near 0, unbounded positive slope

*$k=4$ close to 0 near 0, bounded positive slope

*$k=5$ zero slope

*$k\geq 6$ zero slope to higher order


It seems particularly strange that the density is unbounded for $k=1$ and convex for $k=1,2.$ What is distinct about random normal vectors of length 1 and 2? 
Does anyone have intuition or a statistical experiment that leads one to expect these properties? It would be excellent if the discussion also applied to the Gamma distribution.

More details


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*I understand the formula for the chi-squared distribution and the distribution's construction from normals. 

*I understand that as a normal random vector becomes longer then its norm-squared becomes somehow more dispersed away from 0. However this does not fully explain most of the above properties. 

 A: There are two opposing phenomena going on when you turn $k$ standard Normals into a $\chi^2_k$: squaring and adding.
To look at the adding, consider a $\chi_k$ distribution, the square root of $\chi^2_k$.  It has a bounded density even for $k=1$ (where it's the half-Normal distribution). As $k$ increases, the distribution moves further from zero. At $k=1$ the maximum is at zero, at $k=2$ it's about 1.25.  The distribution is becoming closer to Normal and its coefficient of variation is decreasing, so zero is becoming further and further out in the right tail of the approximating Normal. That's why you get increasingly high-order contact of the pdf at 0.
The un-boundedness at zero is because of the squaring (the $\chi$ distribution doesn't have it).  The density of a half-normal is flat at zero, but squaring transforms the $[0,\varepsilon]$ interval to the much narrower $[0,\varepsilon^2]$. So, if there's a constant density over some small interval relatively close to zero, the probability in $[0,\epsilon]$ will be proportional to $\epsilon$ on the original scale, but on the squared scale the the probability in $[0,\epsilon^2]$ will be proportional to $\varepsilon$, so the density goes up without bound.
