MLE for an exponential rate

Times between trades $X_1, \ldots, X_n$ are i.i.d. Exponential$(\lambda)$ with density $\lambda e^{-\lambda x}$ for $x \ge 0$.

(a) Find the MLE $\hat\lambda$.

(b) Is it unbiased? If not, give an unbiased estimator. (Use that $\sum X_i \sim$ Gamma$(n, \lambda)$.)

(c) Use the Fisher information to give an approximate 95% confidence interval for $\lambda$ when $n = 100$ and $\bar x = 2$ seconds.