Is there any way to calculate harmonic or geometric mean having probability density function?I have probability density of function of some data (it's triangular.) How can I calculate harmonic or geometric mean of the data? I know for calculating arithmetic mean of a variable like K, I have to calculate ∫ 0 ∞ K . P ( K ) d K but I don't have any ideas for other types of averaging methods (Harmonic and geometric).

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Is there any way to calculate harmonic or geometric mean having probability density function?I have probability density of function of some data (it&#039;s triangular.) How can I calculate harmonic or geometric mean of the data? I know for calculating arithmetic mean of a variable like K, I have to calculate       ∫          0        ∞    K  .  P  (  K  )  d  K but I don&#039;t have any ideas for other types of averaging methods (Harmonic and geometric).

kidoceanoe · Accepted Answer

Step 1Geometric mean of the data   (      x    1    ,  …  ,      x    n    ) with       x    i    &amp;gt;  0 is defined as   g  =  (      ∏          i      =      1        n        x    i        )          1              /            n      .Taking logarithm we have   ln  ⁡  g  =      1    n        ∑          i      =      1        n    ln  ⁡      x    i  , the arithmetic mean of the   ln  ⁡      x    i  s.Suppose G is the geometric mean of the random variable X where   P  (  X  &amp;gt;  0  )  =  1. Then analogous to the previous statement you have   ln  ⁡  G  =  E  (  ln  ⁡  X  ), that is,   G  =  exp  ⁡      (    E    (    ln    ⁡    X    )    )  .Step 2For       x    i    ≠  0, harmonic mean is defined as the reciprocal of the arithmetic mean of   (      1          x      1        ,  …  ,      1          x      n        ).Similarly harmonic mean of a random variable X (with   P  (  X  ≠  0  )  =  1) is defined as   H  =      1          E              (                  1          X                )            .

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