1- The binomial distribution N! B(M) = (N-M)!M! describes the probability of placing M identical particles...

 

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1- The binomial distribution N! B(M) = (N-M)!M! describes the probability of placing M identical particles into N equivalent physical sites, with a limit of one particle/site if the probability of a specific site being occupied is p. Using Stirling's formula (in N! N in N-N), show that for N >> M, so that p < <1, the binomial distribution becomes the Poisson distribution P(M) P(M) = M M! Where u = Np 2- Use the binomial distribution Show that; N! B(M) = PqN-M (N-M)!M! - B(M + 1) = B(M) (M) M+1/q 1- The binomial distribution N! B(M) = (N-M)!M! describes the probability of placing M identical particles into N equivalent physical sites, with a limit of one particle/site if the probability of a specific site being occupied is p. Using Stirling's formula (in N! N in N-N), show that for N >> M, so that p < <1, the binomial distribution becomes the Poisson distribution P(M) P(M) = M M! Where u = Np 2- Use the binomial distribution Show that; N! B(M) = PqN-M (N-M)!M! - B(M + 1) = B(M) (M) M+1/q