Relationship among Exponential, Gamma, and Normal distribution

Question

I am studying stochastic processes where I stumbled upon the theorem that says the sum of exponential distributions is gamma distribution. However, from the central limit theorem, we know that sum of a sufficiently large number of IID random variables converges to Normal distribution. So is the following relationship always true?

Sum of IID exponential ≈ gamma ≈ Normal

Answer 1

The sum of $n$ IID exponential$(\lambda)$ variables has a $\gamma(n,\lambda)$ distribution. Given a variable $Y_n$ with this distribution, the variable $\displaystyle \frac{Y_n-n/\lambda}{\sqrt{n}/\lambda}$ tends in distribution to a standard normal.