Question

Four yellow buses and 2 grey buses that could be in any order (with equal probability) are travelling together with the probability of a delay of at least t seconds between any two consecutive busses being P((t,0)) = et Wt > 0, independent of all other delays. What is the expected delay between the grey buses?

Answer 1

It is given that there are total 4 yellow buses and 2 grey buses. Total number of buses will be 6. These buses could be in aby order. Thus, the possible combination in any order will be total possible combinations = 6! = 720 Let the random variable X represents the number of yellow buses in between two grey buses. So, the random variable X takes value 0, 1, 2, 3, 4. Compute the probability of each value. For X =0, P(X =0)=2x COM Compute the probability for X =1. P(X=1)= (2x4)

Compute the probability for X =2. P(X=2)=(2! ***P} - (2!)(3!), 4! 6x5x4 2! Compute the probability for X = 3. P(X =3)– (2!)(2!)(3!), 4C Compute the probability for X =4. P(X =4)= (2!)(4!), 4C 61X4

Since the probability of delay is given by P(t,00)=efor allt 20. The given is the probability density function of exponential with parameter 1. Hence, the expected time of delay of 1 bus is 1. That is, mean is 1. Expected delay between the grey buses is expected delay=1xP(X = 0)+2x P(x = 1)+3x P(X = 2)+4× P(X = 3) +5x P(X = 4) Put the values of the computed probabilities in above expression. 4 1 2 1 expected delay=1x=+2x +3x +4x=+5x= 3155 15 5+8+9+8+5 15 - 15 _35 Hence, the expected delay between the grey buses is - sec.