|Feb29-12, 09:27 AM||#1|
poisson vs binomial process
I might need you guys to help me see how this proces, will be distributed:
Suppose we have a large amount of elements N(≈1012). I'm simulating a system where I for each iteration damage a random element. If an element gets damaged its damagecounter goes up 1.
So say I pick element number 100123 on the first iteration. Its damagecounter will now count +1 whilst the damagecounters of the other elements in the system will be 0.
I want to write an expression for the probability that a specific element acquires k damages in t iterations. Here are my steps in deriving an expression:
1) The probability for a specific element to get damaged must be 1/N.
2) Thus the probability that a specific element gets k damages after t iterations must be:
p(x=k) = K(t,k) * (1/N)k * (1-1/N)t-k
where K(t,k) denotes the binomial coefficient.
Now, I want to find the total probability for an element to have acquired k damages during our t iterations. Thus I multiply by N and end up with:
p(x=k) = N * K(t,k) * (1/N)k * (1-1/N)t-k
Now first of all, I want to know: Would this expression be correct or am I making any mistakes/illegal assumptions in the derivation?
Second, my model appears to follow a poisson distribution, when I plot the amount of elements with damage 1->10 versus time. Does that fit with this theory? I can't quite see if that is a good thing or not?
|Feb29-12, 04:25 PM||#2|
I won't comment on your derivation.
However, in general, Poisson distribution is a good approximation for the binomial for large N and Np fixed.
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