Beta distribution: find the parameter $\alpha$ of $\mathcal{B}e(\alpha,\frac{1}{3})$

Question

I have this variable with beta distribution : $Y \sim \mathcal{B}e(\alpha,\frac{1}{3})$.

I have to find the value of $\alpha$ such as : $P(Y \leq 0.416) =0.2 $

Formally for $\alpha \geq 0$ , $\beta \geq 0$ and $0 \leq y \leq 1$ the CDF function of $Y$ at 0.416 is:

$P(Y \leq 0.416) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} \int_0^{0.416} t^{\alpha-1} (1-t)^{\beta-1} dt=0.2$

I am not sure how to proceed. Thanks for the help in advance!!

Answer 1

Maple gives the CDF for $\mathcal{B}e(\alpha,\beta)$ as $$ {\frac {\Gamma \left( \alpha+\beta \right) {y}^{\alpha} {\mbox{$_2$F$_1$}(\alpha,1-\beta;\,1+\alpha;\,y)}}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) \alpha}} $$ In any case, you need to use numerical methods to solve $F(0.416) = 0.2$. Maple says $\alpha = 0.8563203833$.