I found the following exercise interesting while reading L. Bain's book on introduction to probability and statistics:
Let $X_1, X_2, . . . , X_{100}$ be a random sample from an exponential distribution, $X_i ∼ EXP(1)$, and let $Y = X_1 + X_2 + · · · + X_{100}$.
(a) Give an approximation for P[Y > 110].
(b) If $\overline{X}$ is the sample mean, then approximate $P[1.1 < \overline{X} < 1.2]$.
I know that an exponential variable of parameter $ \alpha $ has the following density function:
\begin{equation*} f(x) = \begin{cases} \alpha^{-1}e^{-x/\alpha} &, \text{ si } x \geq 0. \\ 0 &, \text{ si } x < 0. \end{cases} \end{equation*}
Also, if the sum of $ n $ exponentials of parameter $ \alpha $ is considered, it is a gamma distribution of parameters $ (n, 1 / \alpha) $. The formula for this density is:
\begin{equation*} f(x) = \begin{cases} \alpha^{-n}e^{-x/\alpha}\frac{x^{n-1}}{(n-1)!} &, \text{ si } x \geq 0. \\ 0 &, \text{ si } x < 0. \end{cases} \end{equation*}
However, I have not found a way to solve this problem by estimating the result based on the above concepts. How can I do this? Perhaps other previous concepts are necessary, or is density functions sufficient? I need some help to do this.