I know that the gamma family of distributions are a two-parameter family, but Chi-square only has one parameter.

How is a Chi-square distribution a gamma distribution if it only has one parameter?

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    $\begingroup$ A $\chi^2_\nu$ distribution is a Gamma$(\nu/2,1/2)$ distribution: The scale parameter is always equal to $1/2$. $\endgroup$ – Xi'an Dec 12 '16 at 6:42
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    $\begingroup$ A standard Normal distribution has no parameters. How could it possibly be a Normal distribution, then? $\endgroup$ – whuber Dec 12 '16 at 15:03
  • $\begingroup$ @whuber, normal has mean and variance as parameters, so that seemed clear to me that it was a gamma dist'b'n. $\endgroup$ – makansij Dec 12 '16 at 16:17
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    $\begingroup$ The standard normal has no parameters. $\endgroup$ – whuber Dec 12 '16 at 16:28

Let's start with the p.d.f. of a gamma-distributed random variable $X$, where $\alpha$ is the shape parameter and $\beta$ is the rate parameter (the p.d.f. is a little bit different if $\beta$ is a scale parameter; both parameters are strictly positive):

$$ f_X(x) = \frac{x ^ {\alpha - 1} \beta ^ \alpha e ^ {-\beta x}}{\Gamma(\alpha)} $$

Now let $\alpha = \nu / 2$ and $\beta = 1/2$. After making these substitutions in the equation above, we get

$$ f_X(x) = \frac{x ^ {\frac{\nu}{2} - 1} e ^ {-x / 2}}{\Gamma(\nu / 2) 2 ^ {\nu / 2}}, $$

which you can recognize as the p.d.f. of a chi-square-distributed random variable. Since we fixed $\beta$ as a constant (1/2), we've transformed a 2-parameter random variable into one that depends on only one parameter ($\nu$).


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