Let X ∼ N ( 0 , 1 ) and define W = { 0 X &lt; 0 X X ≥ 0 = X 1 X ≥ 0 How can I calculate E [ X | W ] ?Here's what I have tried so far:I want to identify this random variable for any value w such that W = w. So I first noticed thatAnd now all I have to do is to calculate E [ X | W = 0 ] . Since W is neither a continuous or a discrete time variable, Im not sure how to express the conditioned density. Any help would be appreciated.Thanks in advance.

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Let   X  ∼      N        (    0    ,    1    )   and define  W  =      {                            0                          X          &amp;lt;          0                                      X                          X          ≥          0                          =  X      1          X      ≥      0      How can I calculate       E        [    X          |        W    ]   ?Here&#039;s what I have tried so far:I want to identify this random variable for any value w such that   W  =  w. So I first noticed thatAnd now all I have to do is to calculate       E        [    X          |        W    =    0    ]  . Since   W is neither a continuous or a discrete time variable, Im not sure how to express the conditioned density. Any help would be appreciated.Thanks in advance.

Bruno Hughes · Accepted Answer

W  =  0 iff   X  ≤  0. So   E  (  X      |    W  =  0  )  =  E  (  X      |    X  ≤  0  ). By symmetry of   N  (  0  ,  1  ) we have   E  (  X      |    X  &amp;gt;  0  )  =  E  (  −  X      |    X  &amp;lt;  0  ). Also   E      |    X      |    =  E  (  X      I          X      &amp;gt;      0        )  +  E  (  X      I          X      &amp;lt;      0        ). Combine these to get   E  (  X      |    W  =  0  )  =      1    2    E      |    X      |   and   E      |    X      |    =            2      π

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