##### Contents

Combinatorics

Probability & Statistics

# 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

```

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