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

Name: 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
Uploaded: 2022-02-23T17:22:57+00:00
Description: 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

Question

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

Clare Acosta · Accepted Answer

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      }        .

Why the laplace stieltjes transform of S is given by E[e^(-alpha S)]=1−alpha int_0^1 e^(−alpha y)P(B >= y)dy

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