I know that $X_i (i=1,2,...,n)$ are i.i.d. Pareto r.v. with parameter = $3$ and mean = $2$. I need to calculate $P(X_1 + ... + X_{40} > 90)$ using a normal approximation.
I understand that any linear combinations of $X_i$ would be Pareto distributed as well. But what the normal approximation as to do with it ?
If you have an i.i.d. collection of random variables $X_1,X_2, \ldots$ with mean $\mu$ and variance $\sigma^2$, then $Z_n = \frac{\sum_{i=1}^n (X_i - \mu)}{\sigma \sqrt{n}}$ converges in distribution to a $N(0,1)$ random variable by the central limit theorem. This means that $P(Z_n \leq z) \to P(N(0,1) \leq z)$ as $n \to \infty$ for any real number $z$.
In this case, you can write the desired probability as $P(Z_{40} > c)$ for some value $c$ and then approximate it as $P(N(0,1) >c)$.