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(statistics_random_proba_distribution)=
# The probability distributions

There are several already-implemented statistical laws in {{uranie}}, that can be called marginal 
laws as well, used to described the behaviour of a chosen input variable. They are usually 
characterised by two functions which are intrinsically connected: the PDF (probability density 
function) and CDF (cumulative distribution function). One can recap briefly the definition of 
these two functions for every random variable $X : \Omega \rightarrow \rm I\!R$ :

- PDF: if the random variable X has a density $f_X$, where $f_X$ is a non-negative Lebesgue-integrable 
function, then

```{math}
P\left\{a\leq X \leq b \right\} = \int_{a}^{b} f_{X}(s)ds
```

- CDF: the function $F_{X} : {\rm I\!R} \rightarrow [0,1]$, given by 

```{math}
F_{X}(x)= \int_{-\infty}^{x}f_{X}(s)ds, \; x \in {\rm I\!R}
```

For some of the distributions discussed later on, the parameters provided to define them are not 
limiting the range of their PDF and CDF: these distributions are said to be infinite-based ones.  
It is however possible to set boundaries in order to truncate the span of their possible values. 
One can indeed define an lower bound $L$ and or an upper bound $U$ so that the resulting distribution 
range is not infinite anymore but only in $[L, U]$. This truncation step affects both the PDF and CDF: 
once the boundaries are set, the CDF of these two values are computed to obtain $P_{L}$ (the 
probability to be lower than the lower edge) and $P_{U}$ (the probability to be lower than the upper 
edge). Two new functions, the truncated PDF $f_{X}^{[L,U]}$ and the truncated CDF $F_{X}^{[L,U]}$ 
are simply defined as 

```{math}
f_{X}^{[L,U]}(x)=\frac{f_{X}(x)}{P_{U}-P_{L}}, \;\; F_{X}^{[L,U]}(x) =\frac{F_{X}(x) - P_{L}}{P_{U}-P_{L}}.
```

These steps to produce a truncate distribution are represented in {numref}`statistics_trunc_principle` 
where the original distribution is shown on the left along with the definition of $L$ 
(the blue shaded part) and $U$ (the green shaded part). The right part of the plot is the resulting 
truncated PDF.

{{
    "```{" "figure" "} " + parent_dir +
        "/methodology/statistics/figures/TruncatedPrinciple.png\n"
    ":align: center\n"
    ":name: statistics_trunc_principle\n"
    + figure_scale + "\n"
    "\n"
    "Principle of the truncated PDF generation (right-hand side) from the orginal one (left-hand side).\n"
    "```"
}}


```{include} /../core/dataserver/attribute/introducing_tstochastic_attribute.md                     
```

```{toctree}
proba_distribution/uniform_law.md
proba_distribution/log_uniform_law.md
proba_distribution/triangular_law.md
proba_distribution/log_triangular_law.md
proba_distribution/normal_law.md
proba_distribution/log_normal_law.md
proba_distribution/trapezium_law.md
proba_distribution/uniform_by_parts_law.md
proba_distribution/exp_law.md
proba_distribution/cauchy_law.md
proba_distribution/gumbel_law.md
proba_distribution/weibull_law.md
proba_distribution/beta_law.md
proba_distribution/gen_pareto_law.md
proba_distribution/gamma_law.md
proba_distribution/inv_gamma_law.md
proba_distribution/student_law.md
proba_distribution/gene_normal_law.md
proba_distribution/compose_law.md
```