I am trying to work out the median of the beta function of $\mathrm{B}(1/2,1/6)$. I have been told the answer to this is $0.9510$ but i'm unsure to get there? Is there a simple formula in order to get a median for any values of $\alpha$ and $\beta$ between $0$ and $1$?
Thanks
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$\begingroup$The median of the beta function is found by calculating $x_0$ such that the following equation is satisfied (in other words, the Cumulative Distribution Function (CDF) of the Beta distribution from $0$ to $x_0$ is $0.5$) $$\int_0^{x_0}x^{\alpha-1}(1-x)^{\beta-1}dx=0.5$$ Note that $B(\alpha,\beta)=\int_0^{1}x^{\alpha-1}(1-x)^{\beta-1}dx=1$, so that $x_0$ has an upper bound of $1$.
When $\alpha=1/2$ and $\beta=1/6$, the value of $x$ is found to be approximately $0.9510$, i.e. $$\int_0^{0.9510}x^{-1/2}(1-x)^{-5/6}dx=0.5$$
For values of $\alpha$ and $\beta$ that lie between $0$ and $1$, I do not think there is a simple closed form method of obtaining the median. Instead a variety of numerical methods have to be used - these are implemented in many software packages, including Matlab and GNU Octave, not to mention Wolfram Alpha.
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