Problem -
Suppose that the likelihood of a customer buying a car from a dealership is conditionally dependent on the time since their last purchase in months (Δt), and is measured to be P(Δt) = Prob(sale|Δt). Keep in mind that P(Δt) is not a distribution over Δt and that ∑ΔtP(Δt) ≠ 1. A dealership maintains a database of all customers who currently own a car purchased from the dealership. Every month, the population of customers in the database will be distributed over Δt, and likewise the number of car sales will be drawn from N(Δt), a probability distribution over Δt.
1. Write a formula for N (Δt), the steady state distribution of monthly sales for each value of Δt, if P(Δt) is known.
2. Prove that N (Δt) is a probability distribution with Δt=0∑∞ N(Δt) = 1. Are there any conditions on the values of P(Δt) that must be true in order for this identity to hold?
3. Come up with an algorithm that will invert N(Δt) to yield P(Δt), assuming that you can measure N(Δt) from the full population of data sampled with essentially infinite amounts of data. What other data, such as initial conditions, do you need to perform the inversion exactly?
4. In practice, performing this inversion is complicated by the fact that the empirically measured steady state distribution of monthly sales N-(Δt) is derived from a finitely sampled population of data, and differs slightly from the true distribution of N(t). Can you come up with an algorithm to recover P(Δt) from N-(t) that is robust against the differences between N(Δt) and N-(Δt) that arise from sampling errors? Again, you may assume that if certain data, such as initial conditions or scaling factors, are requisite for the inversion, you have access to that data. Be sure to mention what is is.
5. Test your algorithm with the model:
P(Δt) = (.05/1 + exp(-0 .1 · (Δt - 50)))
where integer-valued Δt ∈ [0, 100]. Using N(Δt) derived from the formula in item 1 above as a true distribution, generate a sample of data points, Δti. Use this "data" to form a empirical distribution N-(Δt) and test your inversion algorithm in item 4 to recover P(Δt).