Variance of a function of a random variable as function of the original variable Suppose $X$ a random variable and $Y=f(X)$ a function of this variable
I know I can write $\mathbb V(Y)$ as $\mathbb E(f(X)^2)-\mathbb E(f(X))^2$
I would like to know if it is possible to write $\mathbb V(Y)$ as a function of $\mathbb V(X)$
e.g : $\mathbb V(Y)=g(\mathbb V(X))$ given the function $f$
In other words, I need to find an expression for the variance of the transformed variable $Y$ in which the variance of the initial variable $X$ appears explicitely, preferably as its parameter.
Details about the problem :

*

*The distribution of $X$ is unknown, and I don't think we need it because we compute variances of the whole population, not a sample of it.

*As it happens, the function $f$ is continuous, strictly decreasing, and positive (somewhat like $f(x)=1/x$ with $x>0$).


The exact formula for the variance of $Y$ requires use of the function $f$ and the full distribution of $X$ (not just its variance).  Nevertheless, while there is no exact formula of the kind you want, you can get approximate formulae using Taylor approximation (also called the "delta method").

To facilitate analysis using the delta method, suppose we let $\mu$, $\sigma^2$, $\gamma$ and $\kappa$ denote the mean, variance, skewness and kurtosis of $X$ respectively, and suppose that $f$ is differentiable up to the required orders for our formulae.  The first-order Taylor approximation to the variance is:
$$\begin{aligned}
\mathbb{V}[f(X)]
&\approx f'(\mu)^2 \cdot \sigma^2. \\[6pt]
\end{aligned}$$
The second-order Taylor approximation (shown in this related question and answer) is:
$$\begin{aligned}
\mathbb{V}[f(X)]
&\approx f'(\mu) (f'(\mu) - \mu f''(\mu)) \sigma^2 \\[6pt]
&\quad - \Big[ f''(\mu)^2 \mu + (f'(\mu) - \mu f''(\mu)) f''(\mu) \Big] \gamma \sigma^3 \\[6pt]
&\quad + \frac{f''(\mu)^2}{4} (\kappa-1) \sigma^4. \\[6pt]
\end{aligned}$$
So, you can get an approximating formula that uses only the mean and variance of $X$ by using the first-order Taylor approximation to the variance.  This is not an exact result, and it is not a particularly good approximation.  If you are willing to also use the skewness and kurtosis of $X$ then you can use the second-order Taylor approximation to the variance.  This is also not an exact result, but it is a reasonable approximation in a wide class of cases.
