M. Marsili, ICTP

This lesson starts showing how to use the random variable theory to estimates **errors** when you perform an experiment by calculating the **variance** of the random variable X and the variance of its mean value <X>. Then the **characteristics functions** **or generating functions** were introduced as the **Fourier transform** of the probability density function. These ** characteristics functions** are very useful because they can be used to find

**moments**of a random variable. Provided that the

*n*

^{th}moment exists, it can be calculated by differentiat

*ing n*times the

**characteristic function.**Two examples were considered the **Binomial distribution** and the **Normal distribution,** for this two cases the **characteristic function** was calculated. Then It was obtained the first moment of the

**Binomial distribution**and it was demonstrated that for the

**Normal distribution**all the odd moments are zero while e generic formula can be obtain for the even moments.

In the following some properties of the **characteristic function** were presented and they were applied to calculate the characteristic function of a sum of random variables in two cases the

**Normal distribution**and the

**Poisson distribution**. it was shown that the sum of normal distributed variables is also a normal variable and the sum of Poisson distributed variables is also a Poisson variable.

It was introduced the **cumulants** ?_{n} of a **probability distribution**, a set of quantities that provide an alternative to the **moments** of the distribution. The **relation between cumulants and moments** for the first moments were obtained and it was shown that for the **Normal distribution** just the first two **cumulants** are different from zero. Finally the proof of the **Central limit theorem** was given.

Calculate the **characteristic function** and the first three moments of the

**Uniform distribution**.

Using the **characteristic function** formalism calculate that the second moment for the

**Binomial distribution**.

Calculate the first three **cumulants** of the **Uniform** and the **Poisson** distributions.