Question is from stat110 HW7 Q5:
The bus company from Blissville decides to start service in Blotchville, sensing a promising business opportunity. Meanwhile, Fred has moved back to Blotchville, inspired by a close reading of I Had Trouble in Getting to Solla Sollew. Now when Fred arrives at the bus stop, either of two independent bus lines may come by (both of which take him home). The Blissville company’s bus arrival times are exactly 10 minutes apart, whereas the time from one Blotchville company bus to the next is Expo(1/10). Fred arrives at a uniformly random time on a certain day. 10 (a) What is the probability that the Blotchville company bus arrives first? Hint: one good way is to use the continuous Law of Total Probability.
(b) What is the CDF of Fred’s waiting time for a bus?
Let U~Unif(0,10) be the arrival time of the next Blissvile Company Bus and X~Expo(1/10) be the arrival time of the next Blotchville company bus(by memoryless property).
Let’s say we have solved part(a) and gotten $P(X<U)=1/e$.
I proceeded to attempt to solve (b) by:
Let W be Fred’s waiting time. Then,
$ \begin{align} P(W<w)&=P(W<w \mid\ X<U)P(X<U)+P(W<w \mid\ U \ge X)P(U \ge X) \\\ &= P(X<w \mid\ X<U)P(X<U) +P(U<w \mid\ U \ge X)P(U \ge X) \\ &= P(X<w)P(X<U) + P(U<w)P(U \ge X) \end{align}$
Intuitively, this made sense to me since I am partitioning the cases into two, the first one being where Blotchville’s bus arrives first and the second one being where Blissville’s bus arrives first or at the same time. Then, just looking at the fist partition, asking if fred’s waiting time is less than w is the same thing as asking whether Blotchville’s bus arrives in less than w since Blissville’s bus arrival is later and thus shouldn’t matter.
But i am guessing that i am missing something that makes it so that the equality of the second line to the third line doesn’t hold, if I continue to solve that, I get a very different answer from the solution. The given solution would be:
$P(W<w)= 1-P(W>w) = 1-P(X>w)P(U>w)$
I understand this solution but i am not sure why mine does not also get me the answer
Edit: nvm, suddenly after typing it out, it started to make sense, they just clearly aren’t equal because $X<w \mid\ X<U$ will require us to only look at the cases where $X<U$, for e.g. if $U<w$, then clearly $X<w$
nvm, suddenly after typing it out, it started to make sense, they just clearly aren’t equal because $X<w \mid\ X<U$ will require us to only look at the cases where $X<U$, for e.g. if $U<w$, then clearly $X<w$