Scamuzzig2

Scamuzzig2

Answered

2022-10-07

Let B be a random variable, and S = min ( 1 , B ). Can you help me see why the laplace stieltjes transform of S is given by
E [ e α S ] = 1 α 0 1 e α y P ( B y ) d y

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Answer & Explanation

Clare Acosta

Clare Acosta

Expert

2022-10-08Added 7 answers

This is the integrated form (with respect to the distribution of B) of the equality
1 e α min { 1 , x } = 0 1 α e α y [ x y ] d y ,
valid for every x 0. To prove the equality holds, note that the RHS is
0 min { 1 , x } α e α y d y = [ e α y ] y = 0 y = min { 1 , x } = 1 e α min { 1 , x } .

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