Why is the expectation of an exponential function: E [ exp ⁡ ( A x ) ] = exp ⁡ ( ( 1 / 2 ) A 2 ) ?

Question

Why is the expectation of an exponential function:      E    [  exp  ⁡  (  A  x  )  ]  =  exp  ⁡  (  (  1      /    2  )      A    2    )    ?

Audrey Russell · Accepted Answer

Let   X  ∼      N    (  0  ,  1  ) and   a  ∈      R  . Then                    E        [        exp        ⁡        (        a        X        )        ]                            =                  ∫                                    R                                                1                      2            π                          exp        ⁡                  (          −                      1            2                                x            2                    )                exp        ⁡        (        a        x        )                          d                x        =                  ∫                                    R                                                1                      2            π                          exp        ⁡                  (          −                      1            2                    (          x          −          a                      )            2                    +                      1            2                                a            2                    )                                                  =        exp        ⁡                  (                      1            2                                a            2                    )                          ∫                                    R                                                1                      2            π                          exp        ⁡                  (          −                      1            2                    (          x          −          a                      )            2                    )                =        exp        ⁡                  (                      1            2                                a            2                    )                    because  x  ↦      1          2      π        exp  ⁡      (    −          1      2        (    x    −    a          )      2        )  is the density of an       N    (  a  ,  1  ) distribution.

Why is the expectation of an exponential function: E[exp(Ax)]=exp((1/2)A^2)?

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New Questions in Calculus 1