Question

Light Bulbs

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?

Answer 1

A red light bulb (X) has been flashing forever, according to a Poisson process with rate r.

A blue bulb (Y) has been flashing forever, according to an independent Poisson process with rate b.

Since, Red and blue are independent. So, by the additive property of the Poisson distribution

X+Y ~ Poisson (r+p)

1. The expected length of the interval that t belongs to is follows an exponential distribution with parameter (r+p)

But event refers to either bulb flashing is

2. the probability that t belongs to an RR interval is

First find the probability of first even before time t when red flashes is considered

Similarly, find the probability of first even after time t when red flashes is considered

Since, the occurrance of events are independent before and after time t.

the probability that t belongs to an RR interval

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

First event is red flash with probability

Second event followed by a blue flash is