Question

2. Every day, you order take-out dinner from the nearby Chinese place and your roommate orders take-out from the shwarma place, which is a little farther away. The Chinese take-out is delivered after an exponential waiting time with mean 10 minutes, whereas the shwarma arrives after an (independent) exponential waiting time with a mean of 20 minutes. Whoever gets their food first always takes pity on the other and shares. (a) Let Sn be the amount of time, over the course of n days, that you and your roommate spend waiting hungrily before some food arrives. Find limn→∞ P[Sn ≤ 7n]. (b) Let Tn be the number of times, over the course of n days, that the Chinese food comes first. Find limn→∞ P[Tn ≤ .6n].

Answer 1

The solution is shown in the following pictures.

Here CLT stands for Central Limit Theorem which states that sum any n many random variables with finite mean and variance follows normal distribution with suitable mean and variance as . You can consult any book for that. Thank you.