One interesting idea here with the Poisson Distribution that always sort of seemed unintuitive to me was that the probability of of lambda events occurring is NOT greater than 50%. For instance, if we look at the line for lambda = 10, we see that the peak probability (at k = 10), is only at about 14%. This seems very strange, and it also seems that as lambda gets larger and larger the probability that lambda events occur gets smaller and smaller. My guess is that this has to do with the fact that as lambda gets higher, the values AROUND lambda (like lambda + 1, lambda - 1, etc.) get probability values that are closer and closer to lambda's probability value (notice how for lambda = 1, the probability at k = 2 drops greatly compared to k = 1), and so because of this, the actual "max" probability for any number of events (in this case, that number with the highest probability is lambda) must be lower and lower if there are many more events with probability similar to that probability, because at the end of the day all of these probabilities must sum up to 1.
kujjwal
Yup, I had pretty similar thoughts about the peak differences being unintuitive until I considered that the curves have to have the same integral of 1 since all the probabilities must sum to 1. One way I also liked to think about it was that if on average lambda=10, or 10 events occur within a specified time interval, then P(X=10) (the peak) relies on a higher number of independent random events occurring, so it logically would make sense that the probability of "more" events occurring would be lower for independently random events, explaining why the peak tends to be lower for higher values of lambda.
Mehvix
@ShonenMind I believe this true as λ→∞; in this case, the distribution of approaches a normal distribution centered/symmetric about λ. The probability of observing more events than the mean of a Gaussian distribution is 50%.
This is why increasing exposure time (increasing avg. # seen photons) decreases noise.
GarciaEricS
It's amazing that computer graphics touches on so many different areas of science, not even computer science. We have shown that geometry, human biology, probability, linear algebra, calculus, and more are all hugely relevant in even relatively elementary graphics. It's really cool to see such a deep field, but a little bit scary in that there is just so much to learn in the field and you progress that it seems like you could never learn it all.
s3kim2018
We can also model the arrival of individual photos on the sensor with an exponential distribution. I guess in a sense, a poisson distribution is accounting for multiple photons arriving within a time (discrete). Exponential distribution accounts for one photon that can arrive in any time (continuous). Just wanted to throw some probability out there for those wondering how to model the arrival of an individual photon.
One interesting idea here with the Poisson Distribution that always sort of seemed unintuitive to me was that the probability of of lambda events occurring is NOT greater than 50%. For instance, if we look at the line for lambda = 10, we see that the peak probability (at k = 10), is only at about 14%. This seems very strange, and it also seems that as lambda gets larger and larger the probability that lambda events occur gets smaller and smaller. My guess is that this has to do with the fact that as lambda gets higher, the values AROUND lambda (like lambda + 1, lambda - 1, etc.) get probability values that are closer and closer to lambda's probability value (notice how for lambda = 1, the probability at k = 2 drops greatly compared to k = 1), and so because of this, the actual "max" probability for any number of events (in this case, that number with the highest probability is lambda) must be lower and lower if there are many more events with probability similar to that probability, because at the end of the day all of these probabilities must sum up to 1.
Yup, I had pretty similar thoughts about the peak differences being unintuitive until I considered that the curves have to have the same integral of 1 since all the probabilities must sum to 1. One way I also liked to think about it was that if on average lambda=10, or 10 events occur within a specified time interval, then P(X=10) (the peak) relies on a higher number of independent random events occurring, so it logically would make sense that the probability of "more" events occurring would be lower for independently random events, explaining why the peak tends to be lower for higher values of lambda.
@ShonenMind I believe this true as λ→∞; in this case, the distribution of approaches a normal distribution centered/symmetric about λ. The probability of observing more events than the mean of a Gaussian distribution is 50%.
This is why increasing exposure time (increasing avg. # seen photons) decreases noise.
It's amazing that computer graphics touches on so many different areas of science, not even computer science. We have shown that geometry, human biology, probability, linear algebra, calculus, and more are all hugely relevant in even relatively elementary graphics. It's really cool to see such a deep field, but a little bit scary in that there is just so much to learn in the field and you progress that it seems like you could never learn it all.
We can also model the arrival of individual photos on the sensor with an exponential distribution. I guess in a sense, a poisson distribution is accounting for multiple photons arriving within a time (discrete). Exponential distribution accounts for one photon that can arrive in any time (continuous). Just wanted to throw some probability out there for those wondering how to model the arrival of an individual photon.