## distributions.Pareto


Single-parameter Pareto: `P(X > x) = (t / x) ** alpha` for `x >= t`,


Usage


``` python
distributions.Pareto()
```


optionally truncated (conditioned on `X < truncation`).


## Parameters


`t: float`  
Threshold; finite and positive.

`alpha: float`  
Pareto alpha; finite and positive.

`truncation: float`  
Truncation point above [t](distributions.Pareto.md#prospicio.distributions.Pareto.t).


## Raises


`ValueError`  
If a parameter is out of range.


## Examples

``` python
>>> from prospicio.distributions import Pareto
>>> p = Pareto(500.0, 2.0)
>>> round(p.layer(4000.0, 1000.0), 9)
```

200.0


## Attributes

| Name | Description |
|----|----|
| [alpha](#alpha) | Pareto alpha. |
| [t](#t) | Threshold [t](distributions.Pareto.md#prospicio.distributions.Pareto.t). |
| [truncation](#truncation) | Truncation point, or `None`. |

------------------------------------------------------------------------


#### alpha


Pareto alpha.


`alpha: float`


------------------------------------------------------------------------


#### t


Threshold [t](distributions.Pareto.md#prospicio.distributions.Pareto.t).


`t: float`


------------------------------------------------------------------------


#### truncation


Truncation point, or `None`.


`truncation: float | None`


## Methods

| Name | Description |
|----|----|
| [cdf()](#cdf) | Distribution function `P(X <= x)`. |
| [fit()](#fit) | Maximum likelihood fit of the alpha to large losses at or above |
| [layer()](#layer) | Expected loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs [attachment](reinsurance.Layer.md#prospicio.reinsurance.Layer.attachment). |
| [layer_second_moment()](#layer_second_moment) | Second moment of the loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs |
| [layer_variance()](#layer_variance) | Variance of the loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs [attachment](reinsurance.Layer.md#prospicio.reinsurance.Layer.attachment). |
| [lev()](#lev) | Limited expected value `E[min(X, limit)]`. |
| [mean()](#mean) | Mean of the distribution (`inf` if it does not exist). |
| [quantile()](#quantile) | Quantile: the smallest [x](pricing.TabulatedCurve.md#prospicio.pricing.TabulatedCurve.x) with `P(X <= x) >= p`. |
| [sample()](#sample) | [n](distributions.Binomial.md#prospicio.distributions.Binomial.n) draws from stream `stream` of the generator keyed by |
| [std()](#std) | Standard deviation of the distribution. |
| [stop_loss()](#stop_loss) | Expected excess over a retention, `E[max(X - retention, 0)]`. |
| [survival()](#survival) | Survival function `P(X > x)`, accurate far into the tail. |
| [variance()](#variance) | Variance of the distribution (`inf` if it does not exist). |

------------------------------------------------------------------------


#### cdf()


Distribution function `P(X <= x)`.


Usage


``` python
cdf(x)
```


##### Parameters


`x: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### fit()


Maximum likelihood fit of the alpha to large losses at or above


Usage


``` python
fit(
    losses,
    t,
    reporting_thresholds=None,
    censored=None,
    weights=None,
    truncation=None
)
```


[t](distributions.Pareto.md#prospicio.distributions.Pareto.t).


##### Parameters


`losses: list of float`  

`t: float`  
Threshold of the fitted Pareto.

`reporting_thresholds: list of float = None`  
Per-loss thresholds below which a loss would not have been reported; raised to [t](distributions.Pareto.md#prospicio.distributions.Pareto.t).

`censored: list of bool = None`  
`True` where a loss was capped by a policy limit.

`weights: list of float = None`  

`truncation: float = None`  


##### Returns


`Pareto`  


##### Examples

``` python
>>> from prospicio.distributions import Pareto
>>> round(Pareto.fit([1500.0, 2500.0, 4000.0], 1000.0).alpha, 6)
```

1.10524

------------------------------------------------------------------------


#### layer()


Expected loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs [attachment](reinsurance.Layer.md#prospicio.reinsurance.Layer.attachment).


Usage


``` python
layer(limit, attachment)
```


##### Parameters


`limit: float`  
`inf` for an unlimited layer.

`attachment: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### layer_second_moment()


Second moment of the loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs


Usage


``` python
layer_second_moment(limit, attachment)
```


[attachment](reinsurance.Layer.md#prospicio.reinsurance.Layer.attachment).


##### Parameters


`limit: float`  

`attachment: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### layer_variance()


Variance of the loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs [attachment](reinsurance.Layer.md#prospicio.reinsurance.Layer.attachment).


Usage


``` python
layer_variance(limit, attachment)
```


##### Parameters


`limit: float`  

`attachment: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### lev()


Limited expected value `E[min(X, limit)]`.


Usage


``` python
lev(limit)
```


##### Parameters


`limit: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### mean()


Mean of the distribution (`inf` if it does not exist).


Usage


``` python
mean()
```


##### Returns


`float`  


------------------------------------------------------------------------


#### quantile()


Quantile: the smallest [x](pricing.TabulatedCurve.md#prospicio.pricing.TabulatedCurve.x) with `P(X <= x) >= p`.


Usage


``` python
quantile(p)
```


##### Parameters


`p: float`  
Probability in `[0, 1]`.


##### Returns


`float`  


##### Raises


`ValueError`  
If [p](models.Elpd.md#prospicio.models.Elpd.p) is outside `[0, 1]`.


------------------------------------------------------------------------


#### sample()


[n](distributions.Binomial.md#prospicio.distributions.Binomial.n) draws from stream `stream` of the generator keyed by


Usage


``` python
sample(n, seed, stream=0)
```


[seed](aggregate.EventSet.md#prospicio.aggregate.EventSet.seed).


##### Parameters


`n: int`  

`seed: int`  

`stream: int = ``0`  


##### Returns


`list of float`  


------------------------------------------------------------------------


#### std()


Standard deviation of the distribution.


Usage


``` python
std()
```


##### Returns


`float`  


------------------------------------------------------------------------


#### stop_loss()


Expected excess over a retention, `E[max(X - retention, 0)]`.


Usage


``` python
stop_loss(retention)
```


##### Parameters


`retention: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### survival()


Survival function `P(X > x)`, accurate far into the tail.


Usage


``` python
survival(x)
```


##### Parameters


`x: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### variance()


Variance of the distribution (`inf` if it does not exist).


Usage


``` python
variance()
```


##### Returns


`float`