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Let $X \sim Beta(a,b)$. I was wondering what is the distribution of $\frac{1}{X}$?. Here is my derivation by using transformation of random variable.

Let $Y = h (X) = \frac{1}{X}$, which implies $h^{-1} (y) = 1/y$ and $h^{'-1} (y) = -1/y^2$

\begin{align} f_Y (y) &= f_X( h^{-1} (y) ) \left|\frac{d}{dy} h^{-1} (y)\right|\\ &= \frac{1}{B(a,b)} (1/y)^{a-1} (1-1/y)^{b-1} \left|-1/y^2 \right| \\ &= \frac{1}{B(a,b)} (1/y)^{a+2-1} (1-1/y)^{b-1} \\ &= \frac{B(a+2,b)}{B(a,b)} \ \frac{1}{B(a+2,b)}(1/y)^{a+2-1} (1-1/y)^{b-1} \\ &\propto \frac{1}{B(a+2,b)}(1/y)^{a+2-1} (1-1/y)^{b-1} \end{align}

Very likely I am wrong on it. But just wondering if I can saying it is still beta distributed? I would appreciate it if someone could pointing out where I am getting wrong. Thanks a lot.

Penn
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    "But just wondering if I can saying it is still beta distributed?" since beta variates are in $[0,1]$, and things in $[0,1]$ map to $[1,\infty]$ under the mapping $x\to\frac{1}{x}$, we can say without any math that it is not beta distributed. – John Madden Jul 25 '23 at 14:06
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    When shifted left by $1,$ to make its support the non-negative numbers, this is a Beta Prime distribution. See https://en.wikipedia.org/wiki/Beta_distribution#Related_distributions . Notice that $1/X-1 = (1-X)/X=Y/(1-Y)$ where $Y=1-X$ has a Beta$(b,a)$ distribution. This latter transformation, which is the operation performed to convert a probability $Y$ into an odds, is a standard way to go from the Beta to the Beta Prime distribution. – whuber Jul 25 '23 at 14:26
  • @JohnMadden thanks a lot. you are correct. I messed up the domain. In the case, the format of the resulted pdf function "looks like" beta density but it is not a beta pdf. – Penn Jul 25 '23 at 16:20
  • @whuber thanks, I have been confused by the definition of beta and beta prime for a long time. I think the distribution of 1/X and 1/X-1 are different even the domain is only shifted by 1 in the latter case? – Penn Jul 25 '23 at 16:22
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    Of course the distributions differ! But they differ in the extremely simple way described by subtracting $1.$ Such an inconsequential change never rises to the level of giving the shifted distribution a new name or conducting any special new studies of its properties. – whuber Jul 25 '23 at 16:29
  • For some other relations with the beta prime, see https://stats.stackexchange.com/questions/554034/distribution-of-the-exponential-of-an-exponentially-distributed-random-variable/554035#554035 – kjetil b halvorsen Jul 25 '23 at 19:59

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