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

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?

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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