- #1
CAF123
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Homework Statement
Consider ##n## independent trials, each of which results in one of the outcomes ##1,...k## with respective probabilities ##p_1,...p_k, \sum_{i=1}^{k} p_i = 1##. (I interpret this summation as just saying mathematically that definitely one of the outcomes has to occur on each trial). Show that if all the ##p_i## are small, then the probability that no trial outcome occurs more than once is a approximately equal to $$exp(-n(n-1)\sum_{i} p_i^2/2)$$
The Attempt at a Solution
So if all the ##p_i## are small, in comparison to ##n##, then I believe I can approximate this to a Poisson RV (that is where the exp comes in). (Correct?) So, first I can write the prob that no trial occurs more than once as $$p_1 (1-p_1)^{n-1} p_2 (1-p_2)^{n-1}...p_k (1-p_k)^{n-1} \cdot k! = k!\,\prod_{i=1}^{k} p_i (1-p_i)^{n-1}$$ I am not really sure where to go from here. I am guessing at some stage, I will need to take the limit that as n tends to infinity, something tends to ##e## Thanks.