Discrete & Continuous Random Variables - Statistics

Example: Waiting for the BusSuppose I arrive at a random point in time during the 30 minute gap between buses.It takes me to another bus stop where I catch a transfer. Again, assume that myarrival there is at a random point of the 30 minute gap between transfer buses.What is the chance that I waste less than 15 minutes waiting for my buses?Assume that my waiting times X 1 and X 2 are independent and X i ∼Uniform[0,30].P(X 1 + X 2 ≤ 15) = 1 2 · P(X 1 ≤ 15 ∩ X 2 ≤ 15) = 1 2 · 12 · 12 = 1 8See illustration on next slide.Modification of different schedule gaps, e.g., 20 minutes for the first busand 30 minutes for the second bus, should be a simple matter of geometry.54