Scamuzzig2

2022-10-07

Let B be a random variable, and $S=\text{min}\left(1,B\right)$. Can you help me see why the laplace stieltjes transform of S is given by
$E\left[{e}^{-\alpha S}\right]=1-\alpha {\int }_{0}^{1}{e}^{-\alpha y}P\left(B\ge y\right)dy$

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Clare Acosta

Expert

This is the integrated form (with respect to the distribution of B) of the equality
$1-{\mathrm{e}}^{-\alpha min\left\{1,x\right\}}={\int }_{0}^{1}\alpha {\mathrm{e}}^{-\alpha y}\phantom{\rule{thinmathspace}{0ex}}\left[x⩾y\right]\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}y,$
valid for every $x⩾0$. To prove the equality holds, note that the RHS is
${\int }_{0}^{min\left\{1,x\right\}}\alpha {\mathrm{e}}^{-\alpha y}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}y={\left[-{\mathrm{e}}^{-\alpha y}\right]}_{y=0}^{y=min\left\{1,x\right\}}=1-{\mathrm{e}}^{-\alpha min\left\{1,x\right\}}.$

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