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 int_{0}^{infty}K. P(K)dK but I don't have any ideas for other types of averaging methods (Harmonic and geometric).

Carsen Patel

Carsen Patel

Open question

2022-08-13

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).

Answer & Explanation

kidoceanoe

kidoceanoe

Beginner2022-08-14Added 15 answers

Step 1
Geometric mean of the data ( x 1 , , x n ) with x i > 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 > 0 ) = 1. Then analogous to the previous statement you have ln G = E ( ln X ), that is, G = exp ( E ( ln X ) ) .
Step 2
For 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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