I have been unable to find resources regarding the family of distributions with pdf $$ f_\gamma(x) = \begin{cases} \gamma x^{\gamma-1}, & \text{if } 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases} $$ for any $\gamma \geq 1$. I do not have an extensive background in probability/statistics, but I feel like such simple distributions should have a name/be categorized somehow. Could someone confirm if this is the case, and indicate how I could go about referencing them to learn more? Thanks

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    $\begingroup$ This is a Beta $B(\gamma,1)$ distribution. $\endgroup$ – Xi'an May 31 '19 at 15:55
  • $\begingroup$ Great, thank you! If you post as an answer, I will accept it $\endgroup$ – FraGrechi May 31 '19 at 15:58
  • $\begingroup$ @Xi'an: do you want to post your comment(s) as an answer? Better to have a short answer than no answer at all. Anyone who has a better answer can post it. $\endgroup$ – S. Kolassa - Reinstate Monica May 31 '19 at 15:59
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    $\begingroup$ This distribution is extensively discussed on this site because (a) it often shows up in textbook or exam questions and (b) for positive integral $\gamma,$ it is the distribution of the largest of $\gamma$ iid Uniform$(0,1)$ variables. A good resource for questions of this nature is en.wikipedia.org/wiki/…, which immediately identifies this distribution for you. $\endgroup$ – whuber May 31 '19 at 16:46

This is a special case of the beta distribution. In general, beta distribution has parameters $(\gamma,\lambda)$ which I'll write as $\text{Beta}(\gamma,\lambda)$. The beta distribution has support on the unit interval. It has density

$$ f(x;\gamma,\lambda)=\frac{\Gamma(\gamma+\lambda)}{\Gamma(\gamma)\Gamma(\lambda)}x^{\gamma-1}(1-x)^{\lambda-1} $$

and we can observe immediately that a $\text{Beta}(\gamma,1)$ distribution has density

$$ \begin{align} f(x;\gamma,1) & = \frac{\Gamma(\gamma+1)}{\Gamma(\gamma)\Gamma(1)} x^{\gamma-1}\\ & = \gamma x^{\gamma-1} \end{align} $$

which applies a gamma function identity.

Note that the beta distribution is restricted to positive $\gamma, \lambda$, not merely $\gamma \ge 1$.

I see that Xi'an pointed this out in comments. I'll delete my answer if he posts his own.

  • $\begingroup$ I'm not sure this sufficiently answers the question. For example, isn't the minimal sufficient statistic for the family of distributions in the posted questions coarser than that of the family of ALL Beta distributions? $\endgroup$ – Michael Hardy May 31 '19 at 18:48
  • $\begingroup$ @MichaelHardy It sounds like you might be able to write a more comprehensive answer. $\endgroup$ – Sycorax says Reinstate Monica May 31 '19 at 18:50
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    $\begingroup$ @MichaelHardy What does that have to do with this question? $\endgroup$ – StubbornAtom May 31 '19 at 19:22

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