Chi-square distribution

From WilmottWiki

The chi-square distribution is a continuous probability distribution which is bounded below and unbounded above. It has two parameters $a\geq 0$ , the location; $\nu$ , an integer, the degrees of freedom. Its probability density function is given by

$\frac{e^{-(x+a)/2}}{2^{\nu/2}}\sum_{i=0}^{\infty}\frac{x^{i-1+\nu/2}a^i}{2^{2i}j!\Gamma(i+\nu/2)},\;\;\;x\geq 0$ ,

where $\Gamma(\cdot)$ is the Gamma function. The chi-square distribution comes from adding up the squares of $\nu$ normally distributed random variables. The chi-square distribution with one degree of freedom is the distribution of the hedging error from an option that is hedged only discretely. It is therefore a very important distribution in option practice, if not option theory.