An ambulance travels back and forth at a constant speed along aroad of length L.

Question

An ambulance travels back and forth at a constant speed along aroad of length L. At a certain moment of time, an accidentoccurs at a point uniformly distributed on the road. [Thatis, the distance of the point from one of the fixed ends of theroad is uniformly distributed over (0, L).] Assuming that theambulance's location at the moment of the accident is alsouniformly distributed, and assuming independence of the variables,compute the distribution of the distance of theambulance from the accident.

Explanation / Answer

Let U be a point selected unformly between (0,1). Thereare 2 ways to represent this distribution. 1. U~Uniform(0,1) 2. U~Beta(r=1,s=1) So a uniform distribution is a special case of beta. If youknow about beta distribuion, then this problem is verysimple. Select 2 points uniformly between (0,1). Let X =min(U1, U2) and Y = max(U1,U2). X~Beta(1,2) Y~Beta(2,1) Let Z = Y-X~Beta(r=1,s=2) f(z) =(r+s-1)!/[(r-1)!(s-1)!]zr-1(1-z)s-1= 2(1-z) In your problem, you need to scale by L. By change ofvariable, z = t/L dz/dt = 1/L f(t) =f(z)*dz/dt=2(1-t/L) (1/L) Hope this helps, Mike Hope this helps, Mike (This explanation may not make any sense unless you arefamiliar with beta distribution and change of variable.)

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.