I have a random variable $M>1$ whose variance and expected value I know (as well as any other moments). So for example, $\text{var}(M) = c$ and $\mathrm{E}(M) = d$. I was wondering whether there was a way to provide a bound (either lower or upper — both would be useful) on $\text{var}(\frac{1}{M})$?
The simplest bound I have (just from the definition of variance) is,
$\text{var}(\frac{1}{M}) \leq \mathrm{E}(\frac{1}{M^2})$.
However this isn't all that useful to me since I can't calculate the RHS. I have tried as a next step using Jensen's inequality, but that also doesn't seem to help because my next step:
$\mathrm{E}(\frac{1}{M^2}) \geq \frac{1}{\mathrm{E} (M^2)}$,
so I get two oppositely signed inequalities! Therefore I can't seem to conclude anything useful.
I'm guessing there is an inequality like Jensen's, but for variances (or higher central moments) but I can't seem to find one.
Does anyone know how I can estimate bounds here?
Edit: suppose I also know that the random variable has only has support for $M>>1$.
Best Answer
Suppose that $X \geq k$ and that $E[X]=m$ for some given real numbers $m, k$ such that $m\geq k>0$. We want to bound $Var(1/X)$. We know by Jensen’s inequality that $1/m \leq E[1/X] \leq 1/k$. Thus:
\begin{align} Var(1/X) &= E\left[\frac{1}{X^2}\right] - E\left[\frac{1}{X}\right]^2\\ &\leq E\left[\frac{1}{X}\frac{1}{k}\right] - E\left[\frac{1}{X}\right]^2 \\ &= E\left[\frac{1}{X}\right]\frac{1}{k} - E\left[\frac{1}{X}\right]^2 \\ &\overset{(a)}{\leq} \max_{c \in [1/m, 1/k]} \left[\frac{c}{k} - c^2\right] \\ &= \left\{ \begin{array}{ll} \frac{1}{4k^2} &\mbox{ if $m \geq 2k$} \\ \frac{1}{mk} - \frac{1}{m^2}& \mbox{ if $m \in [k, 2k]$} \end{array} \right. \end{align} where (a) is justified because $E[1/X] \in [1/m, 1/k]$. Notice that this bound is identical to Henry’s bound when $m \geq 2k$, but is better when $m \in [k, 2k)$.