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For example, at the beginning of doing this problem (http://math.illinoisstate.edu/krzysio/3-6-10-KO-Exercise.pdf), I was thinking of using $\text{Var}(\text{Total loss}) = \text{Var}(N \cdot L)$, where $N = \text{number of hurricanes}$, $L = \text{loss in each hurricane}$.

$\text{Var}(N \cdot L) = \mathbb{E}[N^2 \mid L] - \mathbb{E}[N \mid L]^2$, I have no idea why it would apply to Law of total variance....

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I think you meant for the expression to be $\text{Var}(\sum_{i=1}^{N} L_i)$ where $L_i$ is the loss of the $i^\text{th}$ hurricane and $N$ is the number of hurricanes. Using the total variance formula this would be decomposed as (assuming hurricane losses are i.i.d.),

$$ \begin{align} \mathsf{Var}\left (\sum_{i=1}^{N} L_i \right ) &= \mathsf{Var}\left [ \mathsf{E} \left (\sum_{i=1}^{N} L_i ~\middle\vert~ N \right ) \right] + \mathsf{E} \left [\mathsf{Var} \left (\sum_{i=1}^{N} L_i ~\middle\vert~ N \right ) \right ] \\[1.5ex] &= \mathsf{Var}[N \, \mathsf{E}(L_1)] + \mathsf{E}[ N \, \mathsf{Var}(L_1)] \\[1.5ex] &= \mathsf{E}(L_1)^2 \,\mathsf{Var}(N) + \mathsf{Var}(L_1)\, \mathsf{E}(N) . \end{align} $$

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