Luke owns an Hotel in lake Tahoe. He is planning to perform somem expensive repairs off season, and he needs to decide now $($before the winter season$)$ how much to invest. During the winter season, customers arrive to the hotel following a Poisson process with rate A$"[$customers$/$day$).$ There are two categories of customers: high spending customers and low spending customers.

In order to establish the distribution of the the spending behaviors, Luke studies the data from the previous years of his business. Based on the collected data, Luke established that, at any time the probability that a customer incoming is a high spending customer is P $($t$).$ Luke believes the observation horizon was long enough to be generalizable, i$.$e$...$ sufficient to determine the distribution of the probability that, if a customer arrives at time t$,$ they will be classidied as a high spending customer.

Independently from their arrival behavior, each customer i$,$ i $=1,...,$ N$($t$)$ spends at the hotel a random amount of

money X$,($room fees, food, SPA services, bar, etc.$).$ In particular, X$,$ is distributed as a uniform U $(1200,2280)$ for low spending customers, and U $(3500,5300)$ for high spending customers. Consider that keeping the hotel opened for the winter season has a cost of C$/$day$.$ The winter$$season$$has$120$How do you model the random variable defining the number of high spending customers joining the hotel at time t$,$ which we refer to as Nh $($t$)?$ What is the expected number of customers that are high paying that will join for the coming$$winter$$season$?$