14. Poisson Process I 

Hi where I can find his textbook please?

@@elmrabti amazon

@@nadekang8198 Do you have it?

Well said! But, actually, stirling is not a necessity. Since when n -> infinity, we have (n k) = (n * (n-1) * ... * (n - k + 1)) / k! -> n^k / k!

you dont need stirling appoximation for that

Thanks 🙏🏻

Thanks for that clarification. I wrote on my notes that this part needed some more explanation.

haha

t is a different nature and not a random variable.

@@schobihh2703 could you clarify it a bit more?

@@nguyentin6294 as mentioned in the video. k is a random variable, i.e. the summation over all k is 1. "t" just specifies omega, the set of all k's. it is only a parameter and not an event. Or otherwa around ask yourself if t would be an event what would P(t) be? it simply doesnt make sense to ask this, without specifing how much arrivals we are talking about.

@@schobihh2703 thank you for the insight :)

I doubt that my comment will be useful to you at all since it's been 7 years, but I'll comment anyway for people who are going to view this lecture in the future. The second order terms exist because P(k,d) isn't 0 when k >=2. d here stands for delta. Regardless of how small an interval you pick, it is possible for more than one successes to occur in the interval. That probability, however, will converge to 0 much "faster" than d, so we can ignore those probabilities since they are trivial. This is an important assumption to use the binomial theorem, since the binomial theorem only works when there are two possible outcomes - success and failure. If we allow multiple successes, the theorem cannot be used. Of course, it follows that any distribution which doesn't satisfy the property that the probability of multiple successes in a given interval doesn't converge to 0 "faster" than the interval itself cannot be treated as a Poisson Distribution. In our course notes at University of Waterloo, this property is referred to as "individuality".

@@aryanparikh9085 If possible, can you share the notes of this particular characteristic for qualifying a distribution as Poisson?

the course is applied probability, the point is not rigorous derivation

I would look into a textbook for complete rigorous derivations.

Anyone can stump on something once in a while.

He wrote the textbook which contains the detailed derivation procedures. He would probably be able to figure out how to derive the final Poisson formula if he had spent perhaps some more minutes. But the professor decided to leave the students to read the textbook themselves. By the way, some people do have the ability to memorize all details once they take a look. But that is a painful disease.

Πώς ξέρετε ότι είναι Έλληνας; Αγνοήστε τα λάθη, χρησιμοποιώ λογισμικό μετάφρασης.

@@lekokotonteso Tsitsiklis is a greek surname. Furthermore focus on his accent