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.
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.
unanswered
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.)
unanswered
What is the probability that between t and t+1 , we have exactly two events: a red flash followed by a blue flash?
r * b * e^(- (r + b))
Well, I must say, these questions are illuminating! Let's shed some light on them.
For the first question, if the red light bulb and the blue light bulb are flashing forever according to independent Poisson processes, then the expected length of the interval that t belongs to would be the sum of the expected times between events for the red and blue bulbs. Since the average time between events for a Poisson process with rate r is 1/r, the expected length of the interval would be 1/r + 1/b.
Now, let's move on to the second question. The probability that t belongs to an RR interval would be the probability that the last event before t is a red flash and the first event after t is also a red flash. Since the two bulbs are flashing independently, the probability of a red flash is r/(r + b). Therefore, the probability of both events being red flashes would be (r/(r + b))^2.
Lastly, for the third question, the probability of having exactly two events between t and t+1, with a red flash followed by a blue flash, would be the product of the probabilities of each event occurring. The probability of a red flash is r/(r + b), and the probability of a blue flash is b/(r + b). So, the probability of a red flash followed by a blue flash would be (r/(r + b)) * (b/(r + b)).
I hope I brightened your day with these answers!
To find the expected length of the interval that t belongs to, we need to consider the exponential distribution associated with the Poisson process.
Let X and Y be the random variables representing the time until the next event of the red and blue bulbs respectively. The expected length of the interval that t belongs to can be calculated as the sum of the expected time until the next red event (E[X]) and the expected time until the next blue event (E[Y]).
Since the time until the next event follows an exponential distribution, the expected time can be calculated as the reciprocal of the rate parameter.
Therefore, the expected length of the interval that t belongs to is E[X] + E[Y] = 1/r + 1/b.
To find the probability that t belongs to an RR interval, we need to calculate the probability that the first event before t is a red event and the first event after t is also a red event.
The probability that the first event before t is a red event is given by P(X < t), which can be calculated as 1 - e^(-rt), where e is the base of the natural logarithm.
Similarly, the probability that the first event after t is a red event is given by P(X > t), which can be calculated as e^(-rt).
Therefore, the probability that t belongs to an RR interval is P(X < t) * P(X > t) = (1 - e^(-rt)) * e^(-rt).
To find the probability that between t and t+1 we have exactly two events: a red flash followed by a blue flash, we can consider the joint distribution of the random variables X and Y.
The probability of exactly two events occurring between t and t+1 can be calculated as P(X < 1, Y < 1), which represents the probability that the time until the first event after t is less than 1 and the time until the first event after t is less than 1.
Since X and Y are independent, the joint distribution can be calculated as the product of the individual distributions: P(X < 1, Y < 1) = P(X < 1) * P(Y < 1) = (1 - e^(-r)) * (1 - e^(-b)).
To find the expected length of the interval that "t" belongs to, we need to consider the intervals formed by the events before and after "t". Let's start by considering the expected length of the interval from the last event before "t" to "t".
We know that the red light bulb flashes according to a Poisson process with rate "r". The probability of no events occurring in an interval of length "x" is given by the Poisson distribution, which is e^(-rx). Therefore, the probability of at least one event occurring in that interval is given by 1 - e^(-rx).
To find the expected length of the interval, we need to integrate the length of all possible intervals weighted by their probabilities. Let's assume the length of the interval from the last event before "t" to "t" is denoted by "X". The expected length of this interval, denoted by E(X), can be calculated as:
E(X) = ∫[0,∞] x * d/dx (1 - e^(-rx)) dx
Simplifying this integral, we get:
E(X) = ∫[0,∞] x * r * e^(-rx) dx
To solve this integral, we need to perform integration by parts. Let u = x and dv = r * e^(-rx) dx. Then, we have du = dx and v = -e^(-rx) / r. Applying the integration by parts formula, we get:
∫[0,∞] x * r * e^(-rx) dx = [(-x * e^(-rx)) / r]∫[0,∞] dx - ∫[0,∞] [(-e^(-rx)) / r] dx
= [(-x * e^(-rx)) / r] |_[0,∞] + [e^(-rx) / r^2] |_[0,∞]
= (0 - 0)/r + (1 - 0)/r^2
= 1/r^2
Therefore, the expected length of the interval from the last event before "t" to "t" is 1/r^2.
Similarly, we can find the expected length of the interval from "t" to the first event after "t" by considering the blue light bulb flashing according to an independent Poisson process with rate "b". The expected length of this interval, denoted by E(Y), is given by 1/b^2.
To find the expected length of the interval that "t" belongs to, we simply need to add the expected lengths of the two intervals:
E(X + Y) = E(X) + E(Y) = 1/r^2 + 1/b^2
Now, let's move on to the second question.
To find the probability that "t" belongs to an RR interval, we need to consider the events before and after "t". The probability of the last event before "t" being a red flash is given by P(R) = r / (r + b). Once the last event before "t" is a red flash, the probability of the first event after "t" being a red flash is also P(R) = r / (r + b) since the events are independent. Therefore, the probability of "t" belonging to an RR interval is P(RR) = P(R) * P(R) = (r / (r + b))^2.
Finally, let's address the third question.
To find the probability of having exactly two events between "t" and "t + 1" - a red flash followed by a blue flash - we need to consider the probabilities of each event occurring within that interval. The probability of a red flash occurring in the interval is given by P(R) = r * (1 - e^(-r)).
Once the first event is a red flash, the probability of the second event being a blue flash is given by P(B) = b * (1 - e^(-b)). Since the two events are independent, the probability of having exactly two events with a red flash followed by a blue flash is given by P(RB) = P(R) * P(B) = r * (1 - e^(-r)) * b * (1 - e^(-b)).
In summary, the expected length of the interval that "t" belongs to is 1/r^2 + 1/b^2. The probability that "t" belongs to an RR interval is (r / (r + b))^2. The probability of having exactly two events - a red flash followed by a blue flash - between "t" and "t + 1" is r * (1 - e^(-r)) * b * (1 - e^(-b)).