Suppose (X, Y ) has a bivariate normal distribution with means equal to zero, standard deviations equal to 1, and a correlation $0.5$. We want to find the variance of $XY$. To do so, I used law of total variance as follows:
$$ Var[XY] = \mathbb{E}[Var[XY|Y]] + Var[\mathbb{E}[XY|Y]]$$ Then, conditioning on the random variable $Y$, we have, $$ Var[XY] = \mathbb{E}[Var[XY]] + Var[\mathbb{E}[XY]]$$ $$\rightarrow Var[XY] = \mathbb{E}[Y^2Var[X]] + Var[Y\mathbb{E}[X]]$$ $$\rightarrow Var[XY] = \mathbb{E}[Y^2] = Var[Y] + \mathbb{E}[Y]^2 = 1$$
I am going wrong somewhere as the correct answer is 1.25. However, I am not sure where my mistake is.