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#1




Variance Payment Per Payment
Lets say youre given a 2 parameter Pareto (alpha = 2 theta = 300) with deductible 500
Is there a quick way to calculate variance of payment per payment variable without grinding the integrals? Similarily lets say youre given exponential with theta = 300 and deductible 500? is there a way to calculate variance of payment per paymet 
#2




If X~Pareto(alpha = a, theta = 300) and d=500, then X500X>500 ~Pareto(alpha = a, theta' = 300+500=800).
V(X500X>500)=(800/(a1))^2 * (a/(a2)). Note when a=2, variance of payment per payment isn't defined. If Y~Exponential(theta = 300) and d=500, then Y500Y>500 ~Exponential(theta = 300). V(Y500Y>500)=theta^2=300^2. These are the two nicest distros when it comes to variance of payment per payment variables! Uniform(0, theta) is pretty nice, too: If Z~Uniform(0, theta) and d=500 (theta>500), then Z500Z>500 ~Uniform(0, theta500). V(Z500Z>500)= (theta500)^2/12.
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#3




Pareto per payment random variable is Pareto with theta increased by the deductible. Variance is mean squared x (alpha)/(alpha  2). So the the variance would not exist with alpha = 2.
Exponential for the per payment random variable is the original exponential. 
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