Let we have ξ which has normal distribution N ( a , σ 2 )And also let's define η = e ξ . I want to find E η and D η for this random value.Here's what I've got:I can find E ξ and D ξ because I know that for ξ: p ( x ) = 1 2 π σ 2 e ( x − a ) 2 2 σ 2 and then we can evaluate that E ξ = ∫ − ∞ ∞ x p ( x ) d xSame things for disperson and I've got that D ξ = E ξ 2 − E ξ 2 = ( a 2 + σ 2 ) − a 2 = σ 2 But how can I use it to evaluate E η and D η ?

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Let we have   ξ which has normal distribution   N  (  a  ,      σ    2    )And also let&#039;s define   η  =      e    ξ  . I want to find       E    η   and       D    η   for this random value.Here&#039;s what I&#039;ve got:I can find       E    ξ   and       D    ξ   because I know that for   ξ:  p  (  x  )  =      1          2      π              σ        2                  e                            (          x          −          a                      )            2                                    2                      σ            2                              and then we can evaluate that      E    ξ    =      ∫          −      ∞              ∞        x  p  (  x  )     d  xSame things for disperson and I&#039;ve got that      D    ξ    =      E                  ξ        2              −      E    ξ    2    =  (      a    2    +      σ    2    )  −      a    2    =      σ    2  But how can I use it to evaluate       E    η   and       D    η  ?

ejigaboo8y · Accepted Answer

If   ξ  ∼  N  (  a  ,      σ    2    ), then   η  =      e          ξ       has a log-normal distribution with:  E  (  η  )  =      e          a      +                        σ          2                2              E  (      η    2    )  =      e          2      a      +      2              σ        2              V  a  r  (  η  )  =  E  (      η    2    )  −  E  (  η      )    2    =  E  (  η      )    2    (      e                  σ        2              −  1  )  =      e          2      a      +              σ        2              (      e                  σ        2              −  1  )

Let we have ξ which has normal distribution N ( a , &#x

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