÷I'm reading an italian book about casual process (Probabilità e modelli aleatori of Enzo Orsingher). At pag 105 there's the probability of the stopping time $T_\beta$. $$P\{T_\beta \leq t\}=P\{\max_{0\leq s\leq t} B(s) \geq \beta\}=$$
$$=2P\{B(t) \geq \beta\} = \sqrt{\frac{2}{\pi}} \int_{\frac{\beta}{\sqrt{t}}}^{\infty} \exp\{-\frac{w^2}{2}\}dw$$ Because the book used the last result : $$P\{T_\beta \leq t\}=P\{\max_{0\leq s\leq t} B(s) \geq \beta\}= 2P\{B(t) \geq \beta\}$$
Now I don't undersatnd why $T_\beta$ has the inverse gaussian density $$f_{\beta}(t)= \frac{|\beta|}{\sqrt{2\pi t^3}}\exp\left\{-\frac{\beta^2}{2t}\right\}$$ for $t >0$. Where did it find?