Find PDF of the second (later) bus coming to a stop (two bus coming to stop question)

Question

I am waiting bus at a bus top. In the next 10 minutes, both bus A and bus B are expected to arrive, and their arrival time are independent to each other.

X = arrival time of bus A.

Y = arrival time of bus B. X and Y are continuous variables, and each follows the uniform distribution over [0,10].

I have two questions regarding to this prompt.

First is let Z be the arrival time of the later of the two. How can I find the pdf of Z and the expectation E[Z]?

And second is let Z be the arrival time of the earlier of the two. How can I find the expectation E[Z] ?

Answer 1

Let $Z_1 = \min\{X,Y\}$ and $Z_2 = \max\{X,Y\}$.

• It may be easier to find the CDF of $Z_2$ first. Note that $P(Z_2 \le z) = P(X \le z, Y \le z) = P(X \le z) P(Y \le z)$. Once you have the CDF, it is straightforward to find the PDF and the expectation.
• For the second question, note that $E[Z_1] + E[Z_2] = E[Z_1 + Z_2] = E[X+Y] = E[X] + E[Y]$. So if you know $E[Z_2]$ from the previous question, then you can almost immediately find $E[Z_1]$.