Two independent customers are scheduled to arrive in the afternoon. Their arrival times are uniformly distributed between 2 pm and 8 pm. Compute:
(a) the expected time of the first (earlier) arrival;
(b) the expected time of the last (later) arrival.
Effort 3. I tried a new approach. Called x is the time of latter arrival, y is the time of earlier arrival. So the considered region would be $0 \le x \le 6$, $0 \le y \le 6$, $y \le x$. The "mean" of the region, first I predicted the centroid. I am able to prove it by:
find the cdf $f(x,y) = 1/18$ (because the volumn should be 1, the area of the base is $0.5*6*6=18$)
Get the double integral over the considered region $E(Y)$ = $\int_0^6$$\int_0^x$ $yf(x,y)dydx$ $=$ $\int_0^6$$\int_0^x$ $y/18$ $dydx$ $=$ 2
So I got the answer is 2 (or 2+2=4pm) for question a (similarly, got 4 (or 2+4=6pm) for question b, which forms the centroid of the triangular region). I wonder if my approach is correct or not.
Can anyone give me a hint?