# Probability – Beta Distribution and Wiener Process

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 $$x in mathbb {R} setminus {0; 1 }$$