I for the most part understand why the solution did what it did, however I was curious why my approach was incorrect.

The punchline of the question is "What is the approximate probability that the insurance company wil have claims exceeding the premiums collected."

The premium is set that it is 100 more than the expected value of a claim amount that is exponentially distributed with a mean of 1000. There are 100 claims in total.

So E[X]=1000 for the claim and E[X]=1100 for the premium. I THINK they both have a variance of 1000^2.

So then you apply CLT to both of these and you get 2 random variables, one for the claim and the other for the premium.

Claims: Normal(100,000;sqrt(100*1000^2)), lets call this RV A

Premiums:Normal(110,000;sqrt(100*1000^2)), lets call this RV B

So now from here I thought I should do something along the lines of

P(B-A>0)

E[B-A]=10,000

VAR[B-A]=VAR[A] + VAR[B] => stddev = 10,000*sqrt(2)

then from here use the standard normal to solve

1 - P(Z<(0-10,000)/(10,000*sqrt(2)).

This doesn't yield the solution of B, .159.

Am I doing some calculation wrong or is this just an entirely incorrect way to approach the problem?

The punchline of the question is "What is the approximate probability that the insurance company wil have claims exceeding the premiums collected."

The premium is set that it is 100 more than the expected value of a claim amount that is exponentially distributed with a mean of 1000. There are 100 claims in total.

So E[X]=1000 for the claim and E[X]=1100 for the premium. I THINK they both have a variance of 1000^2.

So then you apply CLT to both of these and you get 2 random variables, one for the claim and the other for the premium.

Claims: Normal(100,000;sqrt(100*1000^2)), lets call this RV A

Premiums:Normal(110,000;sqrt(100*1000^2)), lets call this RV B

So now from here I thought I should do something along the lines of

P(B-A>0)

E[B-A]=10,000

VAR[B-A]=VAR[A] + VAR[B] => stddev = 10,000*sqrt(2)

then from here use the standard normal to solve

1 - P(Z<(0-10,000)/(10,000*sqrt(2)).

This doesn't yield the solution of B, .159.

Am I doing some calculation wrong or is this just an entirely incorrect way to approach the problem?

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