1. ## probability density problem

Hi,

Could someone help me with this problem?

Suppose you board a bus leaving airport at time T1, where T1~uniformly distributed between 4:00pm-6:00pm.Then you catch a train which takes time T2 to your town where T2 is uniformly between 1 and 2 hours. When you arrive at your town, either a taxi is immediately available or you must wait one hour for your taxi. Both events occur with equal likelihood, no matter when you arrive. Let T3 be the time when you get a taxi.

(a) Please give the pdf of T3. (graph or expression)
(b) Given that you arrive in your town before 6:00pm,what is the expected value of T3?

2. Hello,

T=T1+T2 is still a uniform distribution (I'll let you think over which interval).

Then T3=T+G, where G follows a geometric distribution (because it will give the time you get a taxi)

So T3-T ~ G(1/2)

First condition with respect to T and find the pdf of T3|T.
Then use this formula : $f_{T3}=\int f_{T3,T} dT$ and $f_{T3,T}=f_{T3|T}\times f_T$.

Looks a bit complicated, but not that much.