Question: After your complaint about their service, a representative of an insurance company promised to call you "between $7$ and $9$ this evening". Assume that this means that the time T of the call is uniformly distributed in the specified interval.
Assume that you know in advance that the call will last exactly $1$ hour. From $9$ to $9:30$, there is a game show on $TV$ that you wanted to watch. Let $M$ be the amount of time of the show that you miss because of the call. Compute the expected value of $M$.
What I have understood is $P(M | X < 8:00) = 0$, i.e. probability that show will be missed is $0$ when call is received before $8:00$.
If I consider time b/w $8:00$ to $8:30$, then expected value is $(8:30-8:00)/2 = 15.$ Is it the right way to proceed. I don't know what is actual answer.
Best Answer
If the call arrives before 8, M=0
If call arrives in the interval $[8;8:30]$ your M is uniform in $[0;30]$. This happens with probability $p=\frac{1}{4}$
If the call arrives after 8:30 you miss all your TV show. This happens with probability $p=\frac{1}{4}$
Thus
$$E(M)=15\frac{1}{4}+30\frac{1}{4}=11.25$$
You can expect to miss about 11 minutes of your show