Compound variance

Quote:

Loss sizes for an insurance coverage, before taking any deductible into account, are exponentially distributed with mean 50. Coverage is subject to a deductible of 5. Calculate the variance of payments after the deductible, taking into account payments of 0 on losses at or below the deductible.

Wondering why the following reasoning is wrong?



Either a loss is less than the deductible F(5) or greater than the deductible (1-F(5)). Losses less than 5 have a payment of 0. Losses greater than 5 have an avg payment of E(x)-E(x ^ 5).



So by Bernoulli we have:

Var = F(5)(1-F(5))*[E(x)-E(x^5) - 0]^2 = 176.25

Compound variance