**Product of Two Random Variables:** $ \begin{align} \mathrm{Var}(Z) &= \mathbb E[(Z)^2] - \left(\mathbb E[Z]\right)^2 \tag{1} \\[4pt] \mathrm{Var}(XY) &= \mathbb E[(XY)^2] - \left(\mathbb E[XY]\right)^2 \tag{2} \\[4pt] &=\mathbb E[X^2Y^2] - (\mathbb E[X] * \mathbb E[Y])^2 \tag{3} \\[4pt] &=\mathbb E[X^2] *\mathbb E[Y^2] - \mathbb E[X]^2 *\mathbb E[Y]^2 \tag{4} \\[4pt] &=(\mathrm{Var}(X)+\mathbb E [X]^2)*(\mathrm{Var}(Y)+\mathbb E [Y]^2)- \mathbb E[X]^2\mathbb E[Y]^2 \tag{5} \end{align} $ where: - (2) Treating $XY$ as a r.v. $Z$ and applying the [[Variance]] formula to it. - (3) Given the independence between $X \perp Y$we can split up the [[Expectation of a Product]] into separate expectations. - (4) The expectation of the second moment $\mathbb E[X^2] = \mathrm{Var}(X) + \mathbb E[X]^2$. **Product of Multiple Random Variables:** $ \begin{align} \mathrm{Var}(X_1\cdots X_n) &= \mathbb E[(X_1\cdots X_n)^2] - (\mathbb E[X_1\cdots X_n])^2 \\[6pt] &= \mathbb E[X_1^2*\cdots *X_n^2]-(\mathbb E[X_1]*\cdots *\mathbb E[X_n])^2\\[6pt] &= \mathbb E[X_1^2]*\cdots *\mathbb E[X_n^2] - (\mathbb E[X_1])^2 *\cdots *(\mathbb E[X_n])^2\\ &= \prod_{i=1}^n \big(\mathrm{Var}(X_i)+(\mathbb E[X_i])^2\big) - \prod_{i=1}^n (\mathbb E[X_i])^2 \end{align} $