Sorry if it is not an appropriate place for such questions, but anyway can anybody please confirm that the formula for the density function of the four-parameter Beta distribution is correct in Wikipedia. It seems $(c - a)$ is missing in the denominator. Thank you.

Best regards, Ivan


1 Answer 1


Yes, the factor is indeed missing.

Let $X$ be standard 2-parameter Beta random variable. The four-parameter one $Y$ is obtained by affine transformation $Y = (c-a) X + a$ for $c>a$. Then $$ f_Y(y) = \frac{1}{c-a} f_X\left(\frac{y-a}{c-a}\right) = \frac{1}{c-a} \left(\frac{y-a}{c-a} \right)^{\alpha-1} \left(\frac{c-y}{c-a} \right)^{\beta-1} \frac{\mathbf{1}(a < y <c)}{B(\alpha,\beta)} = \frac{(y-a)^{\alpha-1} (c-y)^{\beta-1}}{(c-a)^{\alpha+\beta-1}}\frac{\mathbf{1}(a < y <c)}{B(\alpha,\beta)} $$

  • $\begingroup$ Thanks. If anybody knows how to fix it in Wikipedia, can you please do it? $\endgroup$
    – Ivan
    Commented Aug 6, 2012 at 18:27
  • $\begingroup$ @Ivan: Done. It's straightforward to contribute to Wikipedia, by the way; just click on the "edit" link; you can fix things without registering if you want. $\endgroup$
    – joriki
    Commented Aug 6, 2012 at 18:50
  • $\begingroup$ @joriki : I was about to say all you need to do is click on "edit" and then edit, but I see you beat me to it. I've actually heard professors say they'd like to change things in Wikipedia articles but they don't know how. $\endgroup$ Commented Aug 6, 2012 at 22:18

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