Investigation 14: Normal Rendezvous (due Monday, November 8)

You may work with one other person on this assignment, handing in one report with both names.  Word-processed reports are preferred to hand-written ones.  Please copy/paste Minitab output into a Word file as appropriate.

Reconsider the “random rendezvous” situation analyzed in the day26 handout.  Recall that Dave and Laura arrive independently at a restaurant at a random time that is normally distributed with mean 30 minutes after noon and standard deviation 10 minutes.  Suppose that the first to arrive agrees to wait fifteen minutes for the other to show up.

a) Use Minitab to simulate 1000 arrival times for each person.  Produce a scatterplot of their arrival times, using different symbols for cases in which they meet and cases in which they do not.  [Please be sure that your axes are labeled well.  You can submit the plot that you created in class.]

b) Use this simulation analysis to approximate the probability that they will meet.

c) Repeat a) and b) assuming that their arrival times are independent and normally distributed with mean 30 minutes past noon but with standard deviation 5 minutes.  Also comment on whether the probability is larger or smaller than before, and explain why the answer makes sense.

d) Repeat a) and b) assuming that their arrival times are indpendent and normally distributed with standard deviation 10 minutes, but with Laura's mean at 20 minutes past noon and Dave's mean at 40 minutes past noon.  Also comment on whether the probability is larger or smaller than before, and explain why the answer makes sense.

Let us proceed to a theoretical analysis.  One can show that if X and Y are independent random variables with respective means mu_x and mu_y and respective standard deviations sigma_x and sigma_y, then the random variable F=X-Y has mean E(F)=mu_x-mu_y and variance V(F)=(sigma_x)^2 + (sigma_y)^2.  Furthermore, one can show that if X and Y are both normally distributed, then the difference F also has a normal distribution.  Let the random variable D denote Dave’s arrival time and L denote Laura’s.

e) In the scenario of a) and b), determine the distribution of the difference in their arrival times D-L.  Also draw a sketch of the pdf (either by hand or with the Java applet).

f) Still in the scenario of a) and b), determine the exact probability that Dave and Laura will meet.  [Hint: First determine what has to be true about the random variable F for them to meet.]  Is the exact probability close to the approximation from your simulation?

g) Determine the exact probability that they will meet in the scenario of c).  Is the exact probability close to the approximation from your simulation?

h) Determine the exact probability that they will meet in the scenario of d).  Is the exact probability close to the approximation from your simulation?