need to generate some random data from lognormal distribution, where I set the mode and standart deviation of that lognormal distribution. For this purpose I choose to use random numbers generator from lognormal distribution. This generator takes two numbers, that are mean and sd of underlying normal distribution.So far its clear I need to derive mean and sd of normal distribution, which is underlaying for lognormal distribution where I know mode and sd. I know the equations for derivation of mean and sd:NOTATION: n ( x ) = mean of normal distribution s d ( x ) = s d of normal distribution n ( y ) = mean of lognormal distribution s d ( y ) = sd of lognormal distribution m o d e ( y ) = mode of lognormal distributionEQUATIONS: n ( x ) = 2 ∗ l n ( n ( y ) ) − ( 1 / 2 ) ∗ l n ( s d ( y ) 2 + n ( y ) 2 ) s d ( x ) = − 2 ∗ l n ( n ( y ) ) + l n ( s d ( y ) 2 + n ( y ) 2 ) m o d e ( y ) = e x p ( n ( y ) − s d ( y ) 2 )Here I stuck because I cant get the equation for n ( y ) from these equations, that I need to compute n ( x ). So far I ended: m o d e ( y ) = e x p ( 4 ∗ l n ( n ( y ) ) − 3 / 2 ∗ l n ( n ( y ) 2 − s d ( y ) 2 ) ) m o d e ( y ) 2 / 3 ∗ s d ( y ) 2 = n ( y ) 2 ∗ ( n ( y ) 2 / 3 − m o d e ( y ) 2 / 3 )Can anybody help me to complete this derivation?

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need to generate some random data from lognormal distribution, where I set the mode and standart deviation of that lognormal distribution. For this purpose I choose to use random numbers generator from lognormal distribution. This generator takes two numbers, that are mean and sd of underlying normal distribution.So far its clear I need to derive mean and sd of normal distribution, which is underlaying for lognormal distribution where I know mode and sd. I know the equations for derivation of mean and sd:NOTATION:  n  (  x  )  = mean of normal distribution  s  d  (  x  )  =  s  d of normal distribution  n  (  y  )  = mean of lognormal distribution  s  d  (  y  )  = sd of lognormal distribution  m  o  d  e  (  y  )  = mode of lognormal distributionEQUATIONS:  n  (  x  )  =  2  ∗  l  n  (  n  (  y  )  )  −  (  1  /  2  )  ∗  l  n  (  s  d  (  y      )    2    +  n  (  y      )    2    )  s  d  (  x  )  =  −  2  ∗  l  n  (  n  (  y  )  )  +  l  n  (  s  d  (  y      )    2    +  n  (  y      )    2    )  m  o  d  e  (  y  )  =  e  x  p  (  n  (  y  )  −  s  d  (  y      )    2    )Here I stuck because I cant get the equation for   n  (  y  ) from these equations, that I need to compute   n  (  x  ). So far I ended:  m  o  d  e  (  y  )  =  e  x  p  (  4  ∗  l  n  (  n  (  y  )  )  −  3  /  2  ∗  l  n  (  n  (  y      )    2    −  s  d  (  y      )    2    )  )  m  o  d  e  (  y      )          2          /      3        ∗  s  d  (  y      )    2    =  n  (  y      )    2    ∗  (  n  (  y      )          2          /      3        −  m  o  d  e  (  y      )          2          /      3        )Can anybody help me to complete this derivation?

last99erib · Accepted Answer

If the underlying normal distribution has mean μ and variance       σ    2   then the lognormal has mean   exp  ⁡  (  μ  +      σ    2    /  2  ), mode   exp  ⁡  (  μ  −      σ    2    ), and variance   (  exp  ⁡  (      σ    2    )  −  1  )  exp  ⁡  (  2  μ  +      σ    2    ). You say you set the mode and standard deviation of the lognormal, let&#039;s say as   m and   sThen   m  =  exp  ⁡  (  μ  −      σ    2    ) so       log    e    ⁡  (  m  )  =  μ  −      σ    2   and  μ  =      log    e    ⁡  (  m  )  +      σ    2  leaving us to find   σ. Meanwhile       s    2    =  (  exp  ⁡  (      σ    2    )  −  1  )  exp  ⁡  (  2  μ  +      σ    2    ) which with some substitution and manipulation gives  (  exp  ⁡  (      σ    2    )      )    4    −  (  exp  ⁡  (      σ    2    )      )    3    −                    s        2                    m        2              =  0which is a quartic equation in   exp  ⁡  (      σ    2    ) with what looks like one root greater than   1 when             s      2              m      2        &amp;gt;  0 and so it may be possible to solve for   σ even if unattractive in a closed form.

need to generate some random data from lognormal distribution, where I set the mode and standart dev

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