Accept $ W (t) $ is the standard Viennese process up $ (0; 1) $ and $ {T_x } _ {x in mathbb {R}} $ is a collection of random variables defined by the following relationship:

$$ T_x = mu ( {t in (0; 1) | W (t)> xtW (1) }) $$

Here $ mu $ stands for Lebesgue measure.

Is it true that $ forall x in mathbb {R} exists y (x) in mathbb {R} $ so that $ T_x sim Beta (y (x), y (x)) $?

I know that it is true for $ 0 $ and for $ 1 $ ($ T_0 = mu ( {t in (0; 1) | W (t)> 0 }) $ is known as $ Beta ( frac {1} {2}, frac {1} {2}, frac {1} {2}) $; $ T_1 = mu ( {t in (0; 1) | W (t) – tW (1)> 0 }) $ is known to be distributed evenly $ (0; 1) $ that's the same as $ Beta (1, 1) $). I do not know what to do with it $ x in mathbb {R} setminus {0; 1 } $