Let $X_1,…,X_n$ be i.i.d random variables with expected value $\mu$. Can we compute the expected value of the harmonic mean:
$$
\frac{n}{\sum_{i=1}^{n}\frac{1}{X_i}}
$$
thanks for helpers!
[Math] expected value of harmonic mean
probability
probability
Let $X_1,…,X_n$ be i.i.d random variables with expected value $\mu$. Can we compute the expected value of the harmonic mean:
$$
\frac{n}{\sum_{i=1}^{n}\frac{1}{X_i}}
$$
thanks for helpers!
Best Answer
Certainly not from that information alone. For example, for $n=2$ we have $$ \dfrac{2}{1/X_1 + 1/X_2} = \dfrac{2 X_1 X_2}{X_1 + X_2}$$ If $X_i = \mu$ with probability $1$, the expected value would be $\mu$; if $X_i = \mu/2$ or $3\mu/2$, each with probability $1/2$, it would be $7\mu/16$.
EDIT: Let's assume your random variables are always positive (if not, you may have trouble even defining the expected value of the harmonic mean because of zero or near-zero values of the denominator). The harmonic mean of $n$ positive numbers is always less than or equal to the arithmetic mean, and equal only in the case where all the numbers are equal. So the only way the expected value of the harmonic mean can be $\mu$ is when the random variables are equal to $\mu$ with probability $1$.