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**with mean

**Gaussian**) distribution*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.