Probability & statistics

1.Distribution

A distribution is a set of all relative frequencies 
for all possible events related to a random experiment. 
We talk about probability distribution when the population of 
a sample space  (or the random variable) is discrete; and 
probability density in the case of contnuous variables.

1.1. Discrete distribution

The probability to have an outcome a is fa ; the probability 
to have an outcome b is fb. The probability to have 
an outcome between a and b P(a < X < b) is:


P(a < X < b) = (b - a) x fa
That is the rectangle of area: (b-a) x fa

The mean value = Esperance = s = &sum,p(xi) xi
The variance = dispertion = var =  E(x2) - (E(x))2

1.2. Continue distribution (probability density)

The probability to have an outcome a is f(a); the probability 
to have an outcome b is f(b). The probability to have 
an outcome between a and b P(a < X < c) is :


P(a < X < c) = ∫ f(x) dx  [x: a → c]
That is all the area under the curve betwen a and c


Note that ∫ f(x) dx = 1 	[x: - ∞ → + ∞]

The mean value = Esperance = s = ∫ x f(x) dx
The variance = dispertion = var =  E(x2) - (E(x))2

2. The central limit theorem

The central limit theorem  says that the more the number 
of the random variables is sum the more their distribution becomes 
(converges to) normal (Gaussian distribution).

3. Some models of distribution

In the real world, each phenomena is associated to a model of distribution. 
the most common models are: 

1. The uniform or rectangular ristribution, 
2. The triangular distribution
3. The negative exponential distribution,
4. Binomial distribution
5. Poisson distribution
6. Gaussian or normal distribution