# Table 2 A list of joint distributions

BM

$\mathit{\text{P}}\left(·\leqq \mathit{\text{A}},{\mathit{\text{B}}}_{\theta }\in \mathit{\text{dx}}\right)$

$P\left(\underset{u\leqq \theta }{sup}{B}_{u}\leqq A,{B}_{\theta }\in \mathit{\text{dx}}\right)$

$\left(\frac{\lambda }{2}{e}^{-\lambda |x|}-\frac{\lambda }{2}{e}^{\mathrm{\lambda x}}{e}^{-2\lambda max\left(A,x\right)}\right)\mathit{\text{dx}}$

$P{\left(\underset{u\leqq {g}_{\theta }}{sup}{B}_{u}\leqq A,\phantom{\rule{0.3em}{0ex}}{B}_{\theta }\in \mathit{\text{dx}}\right)}^{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\ast }$

$\frac{\lambda }{2}{e}^{-\lambda |x|}\left(1-{e}^{-2\mathrm{\lambda A}}\right)\mathit{\text{dx}}$

$P\left(\underset{{g}_{\theta }\leqq u\leqq \theta }{sup}{B}_{u}\leqq A,\phantom{\rule{0.3em}{0ex}}{B}_{\theta }\in \mathit{\text{dx}}\right)$

$\frac{\lambda }{2}\frac{1}{1-{e}^{-2\mathrm{\lambda A}}}{1}_{x\leqq A}\left({e}^{-\lambda |x|}-{e}^{\mathrm{\lambda x}-2\mathrm{\lambda A}}\right)\mathit{\text{dx}}$

$P\left(\underset{u\leqq {d}_{\theta }}{sup}{B}_{u}\leqq A,{B}_{\theta }\in \mathit{\text{dx}}\right)$

$\left(1-\frac{{x}^{+}}{A}\right){1}_{x\leqq A}\frac{\lambda }{2}\left({e}^{-\lambda |x|}-{e}^{\mathrm{\lambda x}-2\mathrm{\lambda A}}\right)\mathit{\text{dx}}$

$P\left(\underset{\theta \leqq u\leqq {d}_{\theta }}{sup}{B}_{u}\leqq A,{B}_{\theta }\in \mathit{\text{dx}}\right)$

$\left(1-\frac{{x}^{+}}{A}\right){1}_{x\leqq A}\frac{\lambda }{2}{e}^{-\lambda |x|}\mathit{\text{dx}}$

$P\left(\underset{{g}_{\theta }\leqq u\leqq {d}_{\theta }}{sup}{B}_{u}\leqq A,{B}_{\theta }\in \mathit{\text{dx}}\right)$

$\frac{1-\frac{{x}^{+}}{A}}{1-{e}^{-2\mathrm{\lambda A}}}{1}_{x\leqq A}\frac{\lambda }{2}\left({e}^{-\lambda |x|}-{e}^{\mathrm{\lambda x}-2\mathrm{\lambda A}}\right)\mathit{\text{dx}}$

1. ()Note: We see on this line that $\underset{u\leqq {g}_{\theta }}{sup}{B}_{u}$ and B θ are independent.