Consider a Poisson process with rate λ=4, and let N(t) be the number of arrivals during the time interval [0,t].
Suppose that you have recorded this process in a movie and that you play this movie at twice the speed. The process that you will be seeing in the sped-up movie satisfies the following (pick one of the answers):
a) is a Poisson process with rate 2
b) is a Poisson process with rate 4
c) is a Poisson process with rate 8
d) is not a Poisson process
Correct answer is 8.....i.e option (c)
Well, in a sped-up movie, everything appears to happen faster. So, if the original Poisson process had a rate of λ=4, playing it at twice the speed means that it will now appear as if the arrivals are happening at a faster rate. Therefore, the rate in the sped-up movie would be double the original rate, which is 2λ=8.
So, the answer is c) is a Poisson process with rate 8.
But please don't confuse this with how my humor process works. That's a whole different kind of process!
The correct answer is c) is a Poisson process with rate 8.
Explanation:
A Poisson process is a random process that represents the arrivals of events over time. It has two key properties:
1. The number of events in non-overlapping time intervals are independent.
2. The number of events in a given time interval follows a Poisson distribution.
Given that the original Poisson process has a rate of λ=4, it means that on average, there are 4 arrivals in one unit of time.
When the movie is played at twice the speed, the time interval is effectively halved. However, the rate at which events occur remains the same. Therefore, in the sped-up movie, the process would still be a Poisson process but with a rate that is doubled.
Since λ (rate) in the original process is 4, the rate in the sped-up movie would be 2 times 4, which gives a rate of 8.
Hence, the process in the sped-up movie is a Poisson process with rate 8 (option c).
To determine the rate of the process in the sped-up movie, we need to understand the relationship between the rate parameter of a Poisson process and the speed at which the process is observed.
In a Poisson process, the rate parameter, denoted by λ, represents the average number of events occurring in a unit time interval. If λ = 4, it means that, on average, there are 4 arrivals per unit time.
When the movie is played at twice the speed, the time interval is compressed by a factor of 2. This means that the same amount of time in the original process is now represented in half the time in the movie.
To find the rate in the sped-up movie, we divide the original rate parameter by the compression factor. In this case, λ = 4 is divided by 2, yielding a rate of 2.
Therefore, the answer is:
a) is a Poisson process with rate 2.