It makes a big difference how you write you exponential density.
(a)Suppose we obtain a random sample Y1, Y2, . . . , Yn from an Exponential(μ) distribution. Using the log–likelihood function, find an expression for the maximum likelihood estimator for μ.
(b) Suppose the time interval between arrivals of buses at rush–hour at a bus–stop is thought to follow an Exponential(μ) distribution. The first 10 inter–arrival times (in minutes) are observed to be:
4.2, 3.6, 3.0, 9.0, 1.3, 0.8, 2.7, 2.1, 1.8, 1.5.
Find the maximum likelihood estimate for μ. What assumption have you made about the ten measurements?