I might need you guys to help me see how this proces, will be distributed:(adsbygoogle = window.adsbygoogle || []).push({});

Suppose we have a large amount of elements N(≈10^{12}). 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?

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# Poisson vs binomial process

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