Var(X) is known, how to calculate Var(1/X)?

If I have only $\mathrm{Var}(X)$, how can I calculate $\mathrm{Var}(\frac{1}{X})$?

I do not have any information about the distribution of $X$, so I cannot use transformation, or any other methods which use the probability distribution of $X$.


It is impossible.

Consider a sequence $X_n$ of random variables, where



$$\newcommand{\Var}{\mathrm{Var}}\Var(X_n)=1 \quad \text{for all $n$}$$

But $\Var\left(\frac{1}{X_n}\right)$ approaches zero as $n$ goes to infinity:


This example uses the fact that $\Var(X)$ is invariant under translations of $X$, but $\Var\left(\frac{1}{X}\right)$ is not.

But even if we assume $\mathrm{E}(X)=0$, we can’t compute $\Var\left(\frac{1}{X}\right)$:



$$P(X_n=0)=\frac{1}{n} \quad \text{for $n>0$} $$

Then $\Var(X_n)$ approaches 1 as $n$ goes to infinity, but $\Var\left(\frac{1}{X_n}\right)=\infty$ for all $n$.

Source : Link , Question Author : ARAT , Answer Author : Comp_Warrior

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