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  #1  
Old 01-31-2018, 08:13 PM
JohnTravolski JohnTravolski is offline
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Exclamation 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?
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Old 01-31-2018, 08:45 PM
royevans royevans is offline
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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:

E(X^2) = integral of (X^2)*.2/200 from X= 0 to 200.
X is the claim value by the way

Now, multiply [E(x^2) - E(x)^2]by 400.

Last edited by royevans; 02-01-2018 at 12:28 AM..
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Old 01-31-2018, 08:51 PM
royevans royevans is offline
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Double checked and your variance is also correct.
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Old 01-31-2018, 09:57 PM
JohnTravolski JohnTravolski is offline
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Ah, that's a smart way to calculate it. I never think of things like that. I appreciate the help!
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Old 01-31-2018, 10:45 PM
Colymbosathon ecplecticos's Avatar
Colymbosathon ecplecticos Colymbosathon ecplecticos is offline
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Another approach is to use the conditional variance formula:

Var(S) = E(Var(S|N)) + Var(E(S|N))

To use it, let N be the number of claims and S be the sum of the losses.
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Last edited by Colymbosathon ecplecticos; 01-31-2018 at 11:58 PM.. Reason: dyslexia on both terms (and bad variable names)
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Old 01-31-2018, 11:44 PM
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Quote:
Originally Posted by Colymbosathon ecplecticos View Post
Another approach is to use the conditional variance formula:

Var(X) = E(Var(Y|X)) + Var(E(Y|X))

To use it, let X be the number of claims and Y be the sum of the losses.
Should the left hand side be Var(Y)?
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Old 01-31-2018, 11:45 PM
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Quote:
Originally Posted by JohnTravolski View Post
Ah, that's a smart way to calculate it. I never think of things like that. I appreciate the help!
How did you do it?

Roy's approach is pretty close to "first principles."
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Old 01-31-2018, 11:54 PM
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Vorian Atreides Vorian Atreides is offline
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Quote:
Originally Posted by Marcie View Post
Should the left hand side be Var(Y)?
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Old 01-31-2018, 11:57 PM
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Default oops

I corrected the original post.
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Old 01-31-2018, 11:59 PM
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S & N!
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