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The first algorithm uses independent normal distributions: generate $Z_1,\dots ,Z_{\nu }$ with law $N(0,1)$ and returnFor example, the Box-Muller transform generates two such normal variates from independent uniforms:To verify this, $R^2=-2\log U$ has density $e^{-s/2}/2$ and $2\pi V$ is an independent uniform angle. Converting the resulting polar density to Cartesian coordinates gives $(2\pi {)}^{-1}{e}^{-(z_1^2+z_2^2)/2}$, which factors into two independent normal distributions. Independent pairs therefore give the required sum, discarding a spare coordinate when $\nu$ is odd. Its moment-generating function is $(1-2t{)}^{-\nu /2}$, that of the chi-squared distribution.

A genuinely different algorithm uses exponential-envelope rejection sampling for a chi-squared variable. Put $a=\nu /2\ge 1$. The target and proposal probability density functions areFor $a>1$, differentiating $\log (f/g)$ gives $(a-1)(1/y-1/\nu )$, so the maximum is at $y=\nu$. The envelope constant and normalized density ratio areThey remain valid at $a=1$, where $M=1$ and $f=g$. Repeatedly draw fresh independent $U,V$, put $Y=-\nu \log U$, and accept whenThe right side is nonpositive by $\log u\le u-1$, so this is a legitimate acceptance probability. The joint probability of proposing and accepting in $dy$ is $g(y)f(y)/(Mg(y))dy=f(y)dy/M$; conditional on acceptance the density is exactly $f$. This proves the rejection sampling algorithm, rather than only identifying the target law.

For $\nu =2$ I prefer the second method: it reduces to a single exponential draw with no rejection. The first method is simple and attractive when normal draws are already available or $\nu$ is small. For larger $\nu$, the second method avoids generating $\nu$ normals: its acceptance probability is $1/M$, asymptotic to $\sqrt{2\pi }/(e\sqrt{a})$ by Stirling's formula. Its expected number of proposals grows like $\sqrt{\nu }$, rather than the linear number of normal draws in the first method. Both are exact ideal-uniform algorithms; actual runtime also depends on the implementation.