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I have a non-standard beta distribution in the interval [-0.02 , 0.005] (as opposed to [0,1]).

I know its mean and variance (and thus α and β).

I want to calculate the mean of its truncation to [0 , 0.005].

See the following graph for clarification:

Truncate Non-standard Beta distribution for which I need to calculate the mean

Is it possible to derive an equation for this problem?

Thank you.

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There is a special function called the incomplete Beta function, $$B(x;a,b)=\int_0^x t^{a-1}(1-t)^{b-1}\text{d}t\qquad a,b>0$$ which serves both to normalise the density of a Beta $\mathcal{B}(a,b)$ distribution truncated to $(c,d)$: $$f(x;a,b,c,d)=\dfrac{x^{a-1}(1-x)^{b-1}}{B(d;a,b)-B(c,d)}$$ and to define its mean: $$\mathbb{E}_{a,b,c,d}[X]=\dfrac{B(d;a+1,b)-B(c;a+1,b)}{B(d;a,b)-B(c,d)}$$

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  • $\begingroup$ I see the Incomplete Beta Function requires a,b > 0 . My interval though is [-2%,0.5%]. Am I missing something? $\endgroup$ – Hugo González González Feb 6 at 13:22
  • $\begingroup$ I do not understand what an interval [-2%,0.5%] stands for. $\endgroup$ – Xi'an Feb 6 at 13:27
  • $\begingroup$ Please excuse my lack of clarity. If you look at the X axis of my second graph, that is the interval my Beta distribution is defined around. Then it is truncated at 0. I still call it a Beta distribution since i) it has the exact same shape as the one above, and ii) I understand generalized Beta distributions work with negative intervals $\endgroup$ – Hugo González González Feb 6 at 13:56
  • $\begingroup$ I will reword the question for further clarification. $\endgroup$ – Hugo González González Feb 6 at 15:53
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    $\begingroup$ You have to defined which Beta distribution you want to truncate to $[-0.02,0.005]$ and it cannot be a standard Beta since those have support $(0,1)$. If you truncate a distribution, the support of this distribution must be larger than the truncation region. $\endgroup$ – Xi'an Feb 6 at 15:56

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