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#1




Random Sum of Random Variables
There are 400 individuals, each of whom has a 0.2 chance of incurring a claim. If the distribution of the individual claim amount, given that a claim has occurred, is uniform on the interval [0, 200].
Find the mean and variance of the total claim. I calculated the mean of total claim amount to be 8000 and the variance of the total claim amount to be 906,666.67, but I'm honestly not confident in this answer. I was struggling to justify it using conditional distributions and the iterated rule (which is what I assumed I was supposed to use), and instead just relied on intuition to arrive at those answers. Can somebody explain how to justify them, if they're correct? 
#2




You got the mean correct.
When you have "weird distributions" (i.e. conditional or made up distributions), the trick is to think of variance as E(x^2)  E(x)^2. You already know E(x), so you know E(x)^2. Now calculate E(x^2) and you'll get the variance. Here is a hint: Spoiler: Last edited by royevans; 02012018 at 12:28 AM.. 
#4




Ah, that's a smart way to calculate it. I never think of things like that. I appreciate the help!

#5




Another approach is to use the conditional variance formula:
Var(S) = E(Var(SN)) + Var(E(SN)) To use it, let N be the number of claims and S be the sum of the losses.
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"What do you mean I don't have the prerequisites for this class? I've failed it twice before!" "I think that probably clarifies things pretty good by itself." Last edited by Colymbosathon ecplecticos; 01312018 at 11:58 PM.. Reason: dyslexia on both terms (and bad variable names) 
#8




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#9




oops
I corrected the original post.
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"What do you mean I don't have the prerequisites for this class? I've failed it twice before!" "I think that probably clarifies things pretty good by itself." 
#10




S & N!
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