# chi

• Aug 29th 2009, 10:17 PM
Katina88
chi
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 $\displaystyle W_1^k$:
$\displaystyle P(W_1^k < w^k)$
$\displaystyle P(W_1 < w)$

and find the cumulative distribution function and then the density function? does that work, or am i making things up again?
• Aug 29th 2009, 10:50 PM
CaptainBlack
Quote:

Originally Posted by Katina88
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 $\displaystyle W_1^k$:
$\displaystyle P(W_1^k < w^k)$
$\displaystyle P(W_1 < w)$

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

Just show that the Gamma pdf when $\displaystyle \alpha = n/2$ and $\displaystyle \beta = 1/2$ is the pdf of the $\displaystyle \chi^2$.

CB
• Aug 29th 2009, 11:50 PM
Katina88
huh? do i just sub in $\displaystyle \alpha = k/2$and$\displaystyle \beta=1/2$ into the pdf to get full marks? i don't have to show $\displaystyle \chi^2_1$~Gamma(1/2,1/2) first?
• Aug 30th 2009, 04:19 AM
CaptainBlack
Quote:

Originally Posted by Katina88
huh? do i just sub in $\displaystyle \alpha = k/2$and$\displaystyle \beta=1/2$ into the pdf to get full marks? i don't have to show $\displaystyle \chi^2_1$~Gamma(1/2,1/2) first?

I don't know what you have to do to get full marks, but if you know the density of the $\displaystyle \chi^2$ and the Gamma distributions and can show that with particular (valid) parameters they have the same densities then you have shown that they are in such circumstances the same distribution.

(the acceptability of such a demonstration depends on how your course has defined the distributions and if you know their denstities and how having the same distribution has been defined)

CB