### Statistical Mechanics Lecture 13 of 29

April 25, 2012 by Markus Mueller

M. Mueller, ICTP

In this lesson starting from the binomial distribution it is introduced and proved the de Moivre-Laplace theorem with the help of the Stirling's approximation. This theorem states that the binomial distribution of the number of "successes" in n independent Bernoulli trials with probability p of success on each trial is approximately a normal (or Gaussian) distribution with mean np and standard deviation https://md.ictp.it/admin/media/http://upload.wikimedia.org/wikipedia/en/math/2/f/4/2f4995e761574cf8eb9f957d9f250c96.png" alt="\sqrt{npq}"/>, where q is the probability of failure, if n is very large and some conditions are satisfied. Then using the normal distribution it was demonstrated the integral theorem an approximation to calculate the probability that the number of "successes" are between a and b where a < b getting an expression that is exactly the definition of the Riemann integral.

In the second part of the lesson it was considered a different limit for the binomial distribution were the probability of "successes" p depends on n but pn remains constant and p goes to zero when n approach infinity is always finite. Then it is possible to derive the Poisson distribution, a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.

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