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I am working on a CLT problem and am a bit stuck.

The problem:

N1 = 10,000 with claims X1 ~ Exp() with mean = 100

N2 = 3000 with claims X2 ~ Exp() with mean = 200

N3 = 1000 with claims X3 ~ Exp() with mean 1000

All independent, find the Value at Risk for 99%

What I have done so far:

To find the $lambda$ for each of the exponential distributions from the mean:

E[X] = mean and E[X] = $1overlambda$ for Exp()

So...

X1 ~ Exp($lambda$) = $1over lambda$ = 100 so $lambda$ = .01

X2 ~ Exp($lambda$) = $1over lambda$ = 200 so $lambda$ = .005

X3 ~ Exp($lambda$) = $1over lambda$ = 1000 so $lambda$ = .001

With this...

E[X1] = 100 / E[X2] = 200 / E[X3] = 1000

Var[X1] = $1over (.01)^2$ = 10,000

Var[X2] = $1over (.005)^2$ = 40,000

Var[X3] = $1over (.001)^2$ = 1,000,000

Then...

E[S] = 100(N1) + 200(N2) + 1000(N3) = (100)(10,000) + (200)(3000) + (1000)(1000) = 2,600,000

Var[S] = 10,000(N1) + 40,000(N2) + 1,000,000(N3) = (10,000)(10,000) + (40,000)(3000) + (1,000,000)(1000) = 10,220,000,000

So using the CLT then...

$ S - 2,600,000 over sqrt 10,220,000,000$ $le$ 2.326

Is this the correct approach? These numbers seem a little bit off to me.