Hello StackExchange community,
the following excercise showed that I obviously didn't understand how the MLE works:
"Clients are lining up in a burger restaurant. We record the time $t_{1},...,t_{N}$ in minutes required to serve the $N$ consecutive clients. We distinguish between two types of clients: those that are coming to eat a burger and those coming to eat french fries. Service times for all clients are independentendly drawn from an exponential distributions with rate dependent on whether the clients eats a burger or fries:
$p(t_{i}|\theta) = \theta exp(-\theta t_{i})$ (for a burger)
$q(t_{i}|\theta) = 2 \theta exp(-2 \theta t_{i})$ (for fries)
where $\theta$ is a parameter between 0 and $\infty$ to be learned.
We observe six clients: two burger eaters who stayed 2, and 5 minutes, respectively. The last four customers want to eat fries and were served in 1 minute each."
The exercise it now to state the Likelihood function and solve for $\theta$ which I already did. $P(D|\theta) = 16\theta^6 exp (-15\theta)$. The optimal paramter for $\theta$ is then $\theta = 2/5$.
Question "Give the expected time (according to the learned model) taken to serve the next three customers, if every single one wants to eat fries?."
Please help me with that question and provide some explantion(s) on how you came to this conclusion.