Transformed Random Variable - Compute Assessment Answer

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1. Let U ∼ Uniform(0, 1) and g : (0, 1) → R denote any strictly increasing function. Write the cumulative distribution function of the transformed random variable g(U).

2. The commuter’s dream: a bus line where a bus comes every minute, is never late or early, and where service never decreases nor stops at night. You show up at one of those utopic bus stops at noon, not knowing what is the bus schedule. Let Wbus denote the time you have to wait (in minutes). 

(a) Define an appropriate probability model (feel free to bounce ideas during office hour to make sure you are in the right track). 

(b) Compute the probability that you have to wait the bus for more than 6 seconds. 

(c) Compute E[Wbus] and Var[Wbus]. 

(d) Compute the probability that you have to wait the bus for more than 6 seconds given it did not come in the first 3 seconds. Is this probability smaller, equal or bigger than what you found in 1b?  

3.The taxi industry’s dream: enough cabs so that no matter where you are in the city, the expected number of cabs you see in one minute is one. At noon you start waiting for a cab to hail. Let Wcab denote the time you have to wait (in minutes). 

(a) Define an appropriate probability model (feel free to bounce ideas during office hour to make sure you are in the right track).

 (b) Compute the probability that you have to wait for a cab for more than 6 seconds. 

(c) Compute E[Wcab] and Var[Wcab]. 

(d) Compute the probability that you have to wait for a cab for more than 6 seconds given none come in the first 3 seconds. Is this probability smaller, equal or bigger than what you found in 2b? 

(e) Let Lcab denote how long ago the previous cab came (relative to noon, in minutes). For example if the previous cab came at 11:55, L = 5. What is E[Lcab]? 

(f) (Optional, for bonus marks) Let D denote the time difference between the next cab (relative to noon, in minutes) and the one after (for example if the next cab comes at 12:01 and then the one after at 12:03, D = 2). What is E[D]? Try to provide an intuitive explanation for the discrepancy between E[Wcab + Lcab] and E[D], contrasting with the case of the bus inter-arrivals.  $ City populations, continued A city starts off with a population of 1e6 people. Each year, the population either: 

• stays the same, with probability 2/5; 

• doubles, with probability 3/10; 

• quadruples, with probability 1/5; 

• is divided by a factor of two, with probability 1/10.

Ten year later, what is the probability that there are more than 30e6 people? 

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