To prove that the $\displaystyle \chi^2$ distribution with n degrees of freedom is a gamma distribution with $\displaystyle \alpha = n/2$ and $\displaystyle \beta = 1/2$, do i just do:

the cumulative distribution function of $\displaystyle W_1^k$:

$\displaystyle P(W_1^k < w^k)$

instead of the normal

$\displaystyle P(W_1 < w)$

and find the cumulative distribution function and then the density function? does that work, or am i making things up again?