1. ## probability

According to the timetable the London coach is due to arrive at 2pm however because of the variable traffic conditions the coach is likely to be late. let T denote the actual arrival time and X (in minutes) be T-2pm

X is a random variable with pdf = lamda*e^(-lamda(x-w))
where w and lamda are two given constants

find the probability that the coach is late - so does this mean P(X>0) and how do i calculate this?

also if the coach has not arrived by 2pm what is the probability that the coach arrives in the next 10 mins? is this P(0<x<10

2. Originally Posted by twinkle15
According to the timetable the London coach is due to arrive at 2pm however because of the variable traffic conditions the coach is likely to be late. let T denote the actual arrival time and X (in minutes) be T-2pm

X is a random variable with pdf = lamda*e^(-lamda(x-w))
where w and lamda are two given constants

find the probability that the coach is late - so does this mean P(X>0) and how do i calculate this?

[snip]
The problem with the definitions is what happens when the cab is really really late and doesn't arrive until 3pm. What's X meant to be then .....? Technically it will only make sense if you define a time after 2:59pm as 2:60pm, 2:61pm etc. In practice you can probably take the upper terminal in the integral below to be x = 59 without too much loss in accuracy.

$\displaystyle \Pr(\text{Late}) = \Pr(X > 0) = \int_0^{+\infty} \lambda e^{-\lambda (x - w)} \, dx$.

I also have doubts as to what w is meant to be. I assume that the pdf is zero if x - w < 0 => x < w. The whole question looks dubious to be honest. Can taxis arrive early? What happens if they can ......

Originally Posted by twinkle15
According to the timetable the London coach is due to arrive at 2pm however because of the variable traffic conditions the coach is likely to be late. let T denote the actual arrival time and X (in minutes) be T-2pm

X is a random variable with pdf = lamda*e^(-lamda(x-w))
where w and lamda are two given constants

[snip]

if the coach has not arrived by 2pm what is the probability that the coach arrives in the next 10 mins? is this P(0<x<10
I suppose so.

3. thankyou, can you tell me how to go about doing that integration between 0 and infinity

4. Originally Posted by twinkle15
thankyou, can you tell me how to go about doing that integration between 0 and infinity
Where are you stuck? You're integrating an exponential - you should know how to do that. It's an improper integral - have you been taught how to deal with those?