Consider a Poisson process with rate λ=1. Consider three times that satisfy 0<t1<t2<t3. Let M be the number of arrivals during the interval [0,t2]. Let N be the number of arrivals during the interval [t1,t3]. Is the random variable M+N guaranteed to be Poisson?
answer is No
Well, as a Clown Bot, I don't do guarantees, but I can certainly give you some insight!
In this case, M and N are not independent, as M is counting the arrivals up until t2 and N is counting the arrivals from t1 to t3. So, the random variable M+N is not guaranteed to be Poisson.
If M and N were independent, we could say that the sum of two Poisson random variables is also Poisson. But since their intervals overlap, the arrival times are not independent.
So, in summary, the random variable M+N is not Poisson, but hey, at least life is full of surprises, right?
No, the random variable M+N is not guaranteed to be Poisson.
To determine if M+N is Poisson, we need to check if M and N are independent.
In a Poisson process, the number of arrivals in non-overlapping intervals are independent. However, in this case, the intervals [0, t2] and [t1, t3] overlap.
Hence, M and N are not independent, and as a result, the sum M+N is not guaranteed to follow a Poisson distribution. The distribution of M+N will depend on the specific values of t1, t2, and t3.
To determine whether the random variable M+N is guaranteed to be Poisson, we need to consider some properties of Poisson processes.
A Poisson process is a stochastic process that models events occurring at a constant rate λ over time. It has the following key properties:
1. The number of events occurring in non-overlapping intervals of time are independent.
2. The number of events occurring in disjoint intervals are independent.
In this case, we have a Poisson process with rate λ = 1, and we are considering three specific times, t1, t2, and t3, with 0 < t1 < t2 < t3.
Let's analyze the random variables M and N individually:
1. M: The number of arrivals during the interval [0, t2].
This interval is from the start of the process to time t2. Since it is a valid interval for a Poisson process, the random variable M is guaranteed to follow a Poisson distribution with rate λ*t2.
2. N: The number of arrivals during the interval [t1, t3].
This interval is a disjoint interval that spans from time t1 to time t3. However, it is not the entire process interval [0, t3]. Therefore, the random variable N does not directly follow a Poisson distribution with rate λ*(t3 - t1).
To determine whether the random variable M+N is guaranteed to be Poisson, we need to consider the independence of the number of events occurring in [0, t2] and [t1, t3].
If M and N were independent, then the sum of two independent Poisson random variables with rates λ1 and λ2 would indeed be a Poisson random variable with rate λ1 + λ2.
However, in this case, the intervals [0, t2] and [t1, t3] are not guaranteed to be independent. The number of arrivals during these intervals may be correlated, depending on the specific process and the distributions of arrivals over time. Therefore, we cannot conclude that the random variable M+N is guaranteed to be Poisson.
To determine the distribution of M+N, we would need to analyze the joint distribution of M and N, considering the specific relationship between t1, t2, and t3, as well as any other information about the process.