Walter and carl both often need to travel from location a to location b. walter walks,
and his travel time is normal with mean w minutes and standard deviation σ minutes
(travel time can’t be negative without using a tachyon beam, but assume that w is so
much larger than σ that the chance of a negative travel time is negligible).
carl drives his car, and his travel time is normal with mean c minutes and standard
deviation 2σ minutes (the standard deviation is larger for carl due to variability in traffic
conditions). walter’s travel time is independent of carl’s. on a certain day, walter and
carl leave from location a to location b at the same time.
(a) find the probability that carl arrives first (in terms of φ and the parameters). for
this you can use the important fact that if x1 and x2 are
independent with xi∼n(μi,σ2i), then x1+x2∼n(μ1+μ2,σ21+σ22).
(b) give a fully simplified criterion (not in terms of φ), such that carl has more than
a 50% chance of arriving first if and only if the criterion is satisfied.
(c) walter and carl want to make it to a meeting at location b that is scheduled to
begin w+10 minutes after they depart from location a. give a fully simplified criterion
(not in terms of φ) such that carl is more likely than walter to make it on time for the
meeting if and only if the criterion is satisfied.