$A$ and $B$ are supposed to meet.
$A$ arrives in a randomly chosen (uniform distribution) moment between $2$ and $3$ pm.
$ B$ arrives at $2$ pm with probability equal to $0,5$ and in a randomly chosen (with uniform distribution) moment between $2$ and $3$ pm with probability $0,5$
.
Find the expected value of the amount of time which the person who arrives first spends waiting for the other.
Here are my thoughts:
$A$ has a uniform distribution on $[2,3]$ , but I don't know what the random variable for person $B$ looks like.
We need to express random variable whose values are amounts of time of waiting, using the random variables we find for person $A$ and $B$.
Could you explain to how to do that and tell me if it's a good approach?
Best Answer
The random variable for $B$ has pdf $g(b)=1/2$ over the interval from $t=2$ to $t=3$, together with a point mass of $1/2$ at $t=2$. As you've pointed out, $A$ has pdf $f(a)=1$ on this interval. To compute the expected waiting time, it's convenient to break the calculation up into three disjoint pieces:
The first case contributes $${\rm Pr}(B=2) \int_2^3 (a-2)f(a)\,da = 1/4$$ to the expectation. The second case contributes $$\int_2^3 f(a)\,da \int_2^a (a-b)g(b) \,db=1/12.$$ The third case also contributes $1/12$; I leave that integral to you. So the expected waiting time is $5/12$ hour, or $25$ minutes.