How to prove $\text{var}(X_1X_2)\geq \text{var}(X_1)$ when $E(X_2|X_1)=1$ How to prove that $\text{var}(X_1X_2)\geq \text{var}(X_1)$ when $E(X_2|X_1)=1$ ?
I just try $\text{var}(X_1X_2)=E(X_1^2X_2^2)-E(X_1X_2)^2$ and $\text{var}(X_1)=E(X_1^2)-E(X_1)^2$,
but then I do not know how to proceed.

Using the law of total variance,
$$\operatorname{Var}(X_1X_2)
  =E\underbrace{\operatorname{Var}(X_1X_2|X_1)}_{\ge 0}+\operatorname{Var}E(X_1X_2|X_1)
 \ge\operatorname{Var}(X_1\underbrace{E(X_2|X_1)}_{=1})
=\operatorname{Var}X_1.
$$
