For a &gt; 0 and a r.v. X, is it true that ∫ 0 ∞ a p − 1 P [ X ≥ a ] d ⁡ a = E [ ∫ 0 X a p − 1 d ⁡ a ]? If so, why?

Question

For   a  &amp;gt;  0 and a r.v.   X, is it true that       ∫    0    ∞        a          p      −      1            P    [            X        ≥    a    ]  d  ⁡  a  =      E    [            ∫      0                        X                            a              p        −        1              d    ⁡    a    ]? If so, why?

aniizl · Accepted Answer

∫    0    ∞        a          p      −      1            P    [            X        ≥    a    ]  d  ⁡  a  =      ∫    0    ∞        a          p      −      1            ∫                  R                  I          [      x      ≥      a      ]        d      P    (  x  )  d  ⁡  a and by Fubini&#039;s theorem      ∫    0    ∞        a          p      −      1            ∫                  R                  I          [      x      ≥      a      ]        d      P    (  x  )  d  ⁡  a  =      ∫                  R                  ∫    0    ∞        a          p      −      1            I          [      x      ≥      a      ]        d  ⁡  a  d      P    (  x  )  =      E        [                  ∫        0                              X                                      a                  p          −          1                    d      ⁡      a        ]    .

For a > 0 and a r.v. X , is it true that <msubsup> ∫

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