Suppose that a person $H_{1}$ Arrives at a restaurant between 12:00 and 13:00, and $H_{2}$ Arrives (Independently of $P_{1}$), between 12:00 and 14:00. What's the expected waiting time? Suppose that $P_{1}$ and $P_{2}$ Represents the arrive time of $H_{1}$ and $H_{2}$ respectively. Suppose both distributes uniform.
My idea is to make a variable change such I get the distribution of $|P_{1}-P_{2}|$ and then calculate the expected value but... The idea is not to use (I guess) The expected value (Since I haven't seen that on class yet ) And the second problem is, how could I get $|P_{1}-P_{2}|$ using the variable change? I'm confused.
If the arrival times are uniformly distributed on $12$:$00$ to $13$:$00$ for $H_1$ and $12$:$00$ to $14$:$00$ for $H_2$, then we have that $H_1\sim\mathcal{U}(0,1)$ and $H_2\sim\mathcal{U}(0,2)$. You mention that you have not seen the expected value yet, thus imagine have a line segment, then the "midpoint" of such segment will be equivalent to your expectation, which I believe makes geometrical sense. It then follows that $\mathbb{E}(H_1)=(1+0)/2=1/2$ and $\mathbb{E}(H_2)=(2+0)/2=2/2=1$. In other words, the first person will on average arrive at $12$:$30$ and the other at $13$:$00$.