This is what I don't understand too well (sorry to bother you).
Let's assume, as you wrote, that your device has 1GH/s of computing power.
So what T do you like to choose?
If you choose T=1 sec, then the probabilty to find one nonce in T is the mean, so is lambda. Lambda is 1/4.3=0.23. 23%.
The probability to find two nonces in T, according to Poisson distribution, is ((0.2325^2)*e^-0.2325)/2!=0.021. 2,1%.
Depends on what question you are trying to solve. The question you answered above is "What is the probability of finding k nonces in a period of 1s? And in a period of 2s?"
That was not the question I was answering. My question was "What is the probability of finding k nonces
at the same time?"
I think that, thanks to your patience, now I see how you approached the problem.
I wouldn't say "at the same time": Poisson distribution tells you the probability that an event happens N times in a
fixed time
interval (or in a fixed space interval) when you know that the mean in that period is lambda. So with Poisson you have first to choose an interval, and then go on. It can be as short as a... wavefront, or the clock of your hardware, I don't know how to say it well. You chose a hash computation time as the interval, of course.
The problem is independent of thread execution time, because that is relatively constant. They all end at mostly the same time in a wavefront. The problem is: when they all end, how many have nonces?
Oh, ok! I'm sorry to say that I don't know how hardware and software work, so please forgive me for my past assumptions. :-p
The problem is: "every n clocks, we don't care about n, we have our bunch of exactly X simultaneous hashes, all toghether in a wavefront, given by our X concurrent threads (I hope that "threads" is the right word here). Each thread gives us one and only one hash. Since we know that the probability for a hash to be a nonce is 1/2^32, how many Y nonces (with
0<=Y<=X) are we likely to find in a bunch?"
Well, then Poisson is not the most proper distribution for us: it isn't meant to describe this situation.
AFAIK Poisson is thought to calculate the probability of a number of sequential events in a fixed interval. You are using it to calculate the probability of simultaneous events in... no interval.
Using Poisson completely confused me, and led me to guess a different situation and a different problem. I'm sorry for that.
Of course Binomial is the proper distribution, but you're completely right writing that Poisson gives a very good approximation, as you showed, setting lambda = X/(2^32). So Poisson is numerically right, too, for our case.
I think I got your point, now. Thanks again.
