A red light bulb has been flashing forever, according to a Poisson process with rate r . Similarly, a blue bulb has been flashing forever, , according to an independent Poisson process with rate b . Let us fix t to be 12 o'clock.

Question

1. What is the expected length of the interval that t belongs to? That is, find the expected length of the interval from the last event before t until the first event after t . Here, an event refers to either bulb flashing.

2. What is the probability that t belongs to an RR interval? (That is, the first event before, as well as the first event after time t , are both red flashes.)

3. What is the probability that between t and t+1 , we have exactly two events: a red flash followed by a blue flash?

Answer 1

2/(r+b)

r^2/(r+b)^2

r*b*2*e^(r+b)/(r+b)^4

Answer 2

why 3) isnt r*b/(r+b)^2 ?

Answer 3

which is correct ? and reasoning

Answer 4

Why number 3 is not (rbe^-(r+b))/2 ? I get this other answer doing
(r/(r+b))*(b/(r+b))*(((r+b)^2)*e^-(r+b))/2
P(A/\B) = P(B)*P(A|B)

Answer 5

why isn't the answer to question one:

1/(r+b)

where does the 2 in 2/(r+b) come from

Answer 6

Because you have to think it as 2 Poissons, one from t to the next arrival and one from to to the previous arrival (reverse time). So you sum the expected length of the interval of both.

Answer 7

thanks for your answer. that is weird. is the average time between flashes still (either color) equal to 1/(r+b)?

Answer 8

thanks for your answer. that is weird. is the average time between flashes still equal to 1/(r+b)? but due to the fresh-start property t could occur at the beginning of time?

Answer 9

I have the same answer as youtu for number 3. if you add red/blue as he posted to blue/red + red/red + blue/blue you get the total prob of 2 arrivals, which should be the case.