02q1sol - MASSACHUSETTS INSTITUTE F T ECHNOLOGY O 1

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
MASSACHUSETTS OF TECHNOLOGY INSTITUTE 1.203J/6.281J/13.665J/15.073J/16.76J/ESD.216J Logistical and Transportation P lanning Methods Fall 2002 Quiz 1 Solutions (a)(i) Without loss of generality we can pin down X1 at any fixed point. X2 is still uniformly distributed over the square. Assuming that the police car will always follow a shortest route to the emergency incident, the max possible distance between X1 and X2 is 2 km. The travel distance is thus uniformly distributed between 0 and 2 km. (a)(ii) Following similar logic, the max possible distance is now 4 km. The travel distance is thus uniformly distributed between 0 and 4 km. (b) Let's number the links as shown in Figure 1. There is a chance that the emergency incident will be on any one of the 12 links. Thus if we can determine the conditional pdf for the travel distance from X1 (conditioned to be uniformly distributed on link 7) to X2 for each possible link for X2, we are done. All we do then is add the resulting conditional pdfs, multiplying each by A, the probability of occurrence. Careful inspection of the problem reveals that with regard to computing the conditional travel distance pdf between X1 and X2 there are three sets of links Figure 1: Link Numbering Set 1: 1,3,4,5, 6,8,9, 10, 11
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
MASSACHUSETTS OF TECHNOLOGY INSTITUTE 1.203J/6.281J/13.665J/15.073J/16.76J/ESD.216J Logistical and Transportation P lanning Methods Fall 2002 Consider link 1. Define X2 to be the distance from the right most point of link 1. Then the conditional travel distance, given that X1 is defined to be the distance on link 7 from its left most point and that X2 is on link 1, is (Dl, 7) = X1 + X2 + 1. Let's define V = X1 + X2. Note that XI, X2 - U(0,l) i.i.d. Either by convolution, or by the "never fail" sample space method using cumulative distribution functions, or by recalling problem 2(e)(i) of HW1, we find d t [O, 11 fv(d) = otherwise Now the conditional pdf we want for link 1 is fv(d) "shifted to the right" by one unit of distance. Call this conditional pdf f (Dl,7)(d). Then we have for link 1, f (Dl,7)(d) = fv(d - 1). By inspection we also have f (D3,7)(d) = f (D8,7)(d) = f (Dll, 7)(d) = f (Dl, 7)(d) = fv(d - 1). For the remaining links in Set 1, links 4, 5, 6, 9 and 10 "touch" link 7, so there is no shifting of the pdf by one. That is, there is no intermediate link between them that would add 1.0 km to the travel distance. Hence, f (D4,7)(d) = f (D5,7)(d) = f(D6,7)(d) = f(D9,7)(d) = f(Dl0,7)(d) =
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 7

02q1sol - MASSACHUSETTS INSTITUTE F T ECHNOLOGY O 1

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online