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.
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?
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)
2/(r+b)
r^2/(r+b)^2
r*b*2*e^(r+b)/(r+b)^4
why isn't the answer to question one:
1/(r+b)
where does the 2 in 2/(r+b) come from
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.
thanks for your answer. that is weird. is the average time between flashes still (either color) equal to 1/(r+b)?
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?
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.
To answer these questions, we need to utilize the properties of Poisson processes. Let's go through each question step by step.
1. Expected length of the interval that t belongs to:
In a Poisson process with rate r, the time between consecutive events follows an exponential distribution with mean 1/r. Similarly, in a Poisson process with rate b, the time between consecutive events follows an exponential distribution with mean 1/b.
Since the two bulbs have independent processes, the expected length of the interval that t belongs to can be calculated as the sum of the expected times for the last event before t and the first event after t.
Let's denote X as the exponential random variable for the time to the last event before t (for a red bulb) and Y as the exponential random variable for the time to the first event after t (for a red bulb). The expected length of the interval can be computed as E[X] + E[Y].
The expected value of an exponential random variable with rate parameter λ is given by 1/λ. Therefore, the expected length of the interval that t belongs to is:
E[X] + E[Y] = 1/r + 1/r = 2/r
So, the expected length of the interval is 2/r.
2. Probability that t belongs to an RR interval:
An RR interval refers to the event before t as well as the event after t being red flashes.
To determine this probability, we need to consider the probability of a red flash happening before t and the probability of a red flash happening after t.
The probability of a red flash happening before t can be computed as r/(r+b) because the rate r represents the proportion of red flashes in the combined rate of red and blue flashes.
Similarly, the probability of a red flash happening after t is also r/(r+b) because the rate r represents the proportion of red flashes.
Since these two events are independent (the processes are independent), we multiply the probabilities to get the probability that both events occur.
Therefore, the probability that t belongs to an RR interval is (r/(r+b)) * (r/(r+b)) = r^2 / (r+b)^2.
3. Probability of exactly two events between t and t+1: a red flash followed by a blue flash:
To calculate this probability, we need to consider the probability of a red flash occurring in the interval (t, t+1) and the probability of a blue flash occurring immediately after that.
The probability of a red flash happening in the interval (t, t+1) can be computed as r multiplied by the length of the interval (which is 1 hour since we fixed t at 12 o'clock).
The probability of a blue flash immediately following the red flash can be computed as b, the rate of the blue process.
Again, we multiply the probabilities of these two events as they are independent processes.
Therefore, the probability of exactly two events between t and t+1: a red flash followed by a blue flash, is r * b.
I hope this explanation helps you understand how to calculate these probabilities based on Poisson processes.