Variance comparison of $XY$ and $\min(X,Y)$ There are two independent random variables and each is Gaussian with $\mathrm{E}(X)=mx, \mathrm{E}(Y)=my$, variance of X, i.e. $\mathrm{Var}(X)=vx, \mathrm{Var}(Y)=vy$, respectively. I would like to compare the magnitudes of variances of two functions of these random variables, i.e., $X\times Y$ and $\min(X,Y)$.
I tried hard to compute the variances of these two functions of random variables, and failed. I just need to show variance of $XY$ is greater than or equal to the variance of $\min(X,Y)$, i.e., $\mathrm{Var}(XY) \geq \mathrm{Var}(\min(X,Y))$.
I would really appreciate it. Thanks. 
 A: It is good that you have not succeeded, because the relation is not true: sometimes the variance of $\min(X,Y)$ exceeds the variance of $XY.$
Consider what happens to these two variances when any two random variables $X$ and $Y$ are simultaneously multiplied by a positive number $\lambda:$ because $XY$ is multiplied by $\lambda^2,$ its variance is multiplied by $(\lambda^2)^2 = \lambda^4,$ but because the smaller of $X$ and $Y$ is multiplied by $\lambda,$ its variance is multiplied by $\lambda^2.$  Thus, assuming the ratio
$$f_{X,Y} = \frac{\operatorname{Var}(\min(X,Y))}{\operatorname{Var}(XY)}$$
exists, it is multiplied by $\lambda^{-2}.$
You are hoping to show that $f_{X,Y} \lt 1$ no matter what (Gaussian) distributions $X$ and $Y$ might have.
By selecting $\lambda \lt \sqrt{f_{X,Y}},$ chosen to assure $\lambda^{-2} \gt 1/f_{X,Y},$ you will change the ratio to
$$f_{\lambda X, \lambda Y} = \frac{\operatorname{Var}(\min(\lambda X,\lambda Y))}{\operatorname{Var}(\lambda^2XY)} = \lambda^{-2}f_{X,Y} \gt \left(\frac{1}{f_{X,Y}}\right) f_{X,Y} = 1.$$
The pair $\lambda X, \lambda Y$ provides a counterexample to what you want to prove provided $f_{X,Y}$ is nonzero, because the new variables $\lambda X$ and $\lambda Y$ are still independent with Gaussian distributions.

Notice that the only assumption about the random variables $(X,Y)$ needed to construct this counterexample was that $\operatorname{Var}(XY)\ne 0$ and $\operatorname{Var}(\min(X,Y))\ne 0.$  Both of these are (obviously) the case when $(X,Y)$ has a nondegenerate bivariate Gaussian distribution. 
