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 Short-Term Actuarial Math Old Exam C Forum

#1
 RockOn Member SOA Join Date: Dec 2013 Studying for Exam C Favorite beer: Polygamy Porter Posts: 83 SOA #229

Hello,

I'm having trouble understanding the SOA solution for this problem.
To get the expected value of the second derivative with respect to X. Why
Is only the second term multiplied by the density function? Why not 1/theta squared as well?

Thank you.
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#2 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,716 Quote:
 Originally Posted by RockOn Hello, I'm having trouble understanding the SOA solution for this problem. To get the expected value of the second derivative with respect to X. Why Is only the second term multiplied by the density function? Why not 1/theta squared as well? Thank you.
First of all, it's the second derivative with respect to theta, not X.

Second, you're after the expected value of that second derivative as X varies randomly. But the expected value of 1 / theta^2 equals 1 / theta^2, so there's no need to bother multiplying by the density and integrating. But if you do, you'll get the same answer.

By the way, my guess is that such a question is very rare. Prepare for what is typically tested.
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#3
 RockOn Member SOA Join Date: Dec 2013 Studying for Exam C Favorite beer: Polygamy Porter Posts: 83 Quote:
 Originally Posted by Jim Daniel First of all, it's the second derivative with respect to theta, not X. Second, you're after the expected value of that second derivative as X varies randomly. But the expected value of 1 / theta^2 equals 1 / theta^2, so there's no need to bother multiplying by the density and integrating. But if you do, you'll get the same answer. By the way, my guess is that such a question is very rare. Prepare for what is typically tested.
Thank you for the reply Prof Daniel.
I don't think I've asked my question correctly...
I agree that it is the second derivative with respect to theta, not x (I mistyped before).

When getting the expected value of this, why is the first term multiplied by theta squared and the second term multiplied by the given density function? I'm just trying to understand conceptually why one term is multiplied by one thing and the other is multiplied by the given density function and then integrated. And yes, I agree that the integral of 1/theta sq multiplied by theta is 1/theta squared.

In other words, I think I missing a concept with how to get the expected value.
Thank you!
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#4
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,718 Quote:
 Originally Posted by RockOn Thank you for the reply Prof Daniel. I don't think I've asked my question correctly... I agree that it is the second derivative with respect to theta, not x (I mistyped before). When getting the expected value of this, why is the first term multiplied by theta squared and the second term multiplied by the given density function? I'm just trying to understand conceptually why one term is multiplied by one thing and the other is multiplied by the given density function and then integrated. And yes, I agree that the integral of 1/theta sq multiplied by theta is 1/theta squared. In other words, I think I missing a concept with how to get the expected value. Thank you!
You are taking the expected value with respect to X. If a term doesn't have an x in it, it is a constant and there is no need to multiply by the density and integrate.
#5
 RockOn Member SOA Join Date: Dec 2013 Studying for Exam C Favorite beer: Polygamy Porter Posts: 83 Quote:
 Originally Posted by Academic Actuary You are taking the expected value with respect to X. If a term doesn't have an x in it, it is a constant and there is no need to multiply by the density and integrate.

Let me ask it this way: If I am trying to get the expected value of the function, if I multiply the term (1/theta squared) by the given density function (theta/(theta + x)^2) I don't get 1/(theta squared) as a result for this term.

So I guess I'm back to my original question of why is the first term multiplied by theta squared and the second term multiplied by the given density.

Or asked another way, what are you multiplying the first term by in order to get 1/theta squared?

Thank you.
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#6
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,718 Quote:
 Originally Posted by RockOn Hi Sorry, I'm still confused about this. Let me ask it this way: If I am trying to get the expected value of the function, if I multiply the term (1/theta squared) by the given density function (theta/(theta + x)^2) I don't get 1/(theta squared) as a result for this term.
You then integrate over x. The pdf integrates to one. 1/theta squared is a multiplicative constant.
#7
 RockOn Member SOA Join Date: Dec 2013 Studying for Exam C Favorite beer: Polygamy Porter Posts: 83 Quote:
 Originally Posted by Academic Actuary You then integrate over x. The pdf integrates to one. 1/theta squared is a multiplicative constant.
Got it now! Apologies for going 'round in circles on this one. I hope I didn't frustrate anyone! I kept thinking that the first term was being multiplied by theta squared - but I now realize (after both of your explanations) that it too is being multiplied by the given density function.

Thank you for your patience! I do understand this now __________________
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