Note

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# Transform a distribution¶

In this example we are going to use distribution algebra and distribution transformation via functions.

```
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
```

We define some (classical) distributions :

```
distribution1 = ot.Uniform(0.0, 1.0)
distribution2 = ot.Uniform(0.0, 2.0)
distribution3 = ot.WeibullMin(1.5, 2.0)
```

## Sum & difference of distributions¶

It is easy to compute the sum of distributions. For example:

```
distribution = distribution1 + distribution2
print(distribution)
graph = distribution.drawPDF()
view = viewer.View(graph)
```

```
Trapezoidal(a = 0, b = 1, c = 2, d = 3)
```

We might also use subtraction even with scalar values:

```
distribution = 3.0 - distribution3
print(distribution)
graph = distribution.drawPDF()
view = viewer.View(graph)
```

```
RandomMixture(3 - WeibullMin(beta = 1.5, alpha = 2, gamma = 0))
```

## Product & inverse¶

We might also compute the product of two (or more) distributions. For example:

```
distribution = distribution1 * distribution2
print(distribution)
graph = distribution.drawPDF()
view = viewer.View(graph)
```

```
ProductDistribution(Uniform(a = 0, b = 1) * Uniform(a = 0, b = 2))
```

We could also inverse a distribution :

```
distribution = 1 / distribution1
print(distribution)
graph = distribution.drawPDF()
view = viewer.View(graph)
```

```
CompositeDistribution=f(Uniform(a = 0, b = 1)) with f=[x]->[1.0 / x]
```

Or compute a ratio distribution :

```
ratio = distribution2 / distribution1
print(ratio)
graph = ratio.drawPDF()
view = viewer.View(graph)
```

```
ProductDistribution(Uniform(a = 0, b = 2) * CompositeDistribution=f(Uniform(a = 0, b = 1)) with f=[x]->[1.0 / x])
```

## Transformation using functions¶

The library provides methods to get the full distributions of f(x) where f can be equal to:

sin,

asin,

cos,

acos,

tan,

atan,

sinh,

asinh,

cosh,

acosh,

tanh,

atanh,

sqr (for square),

inverse,

sqrt,

exp,

log/ln,

abs,

cbrt.

For example for the usual log transformation:

```
graph = distribution1.log().drawPDF()
view = viewer.View(graph)
```

And for the log2 function :

```
f = ot.SymbolicFunction(["x"], ["log2(x)"])
f.setDescription(["X", "ln(X)"])
graph = ot.CompositeDistribution(f, distribution1).drawPDF()
view = viewer.View(graph)
```

If one wants a specific method, user might rely on the `CompositeDistribution`

class.

```
# Create a composite distribution
# -------------------------------
#
# In this paragraph we create a distribution defined as the push-forward distribution of a scalar distribution by a transformation.
#
# If we note :math:`\mathcal{L}_0` a scalar distribution, :math:`f: \mathbb{R} \rightarrow \mathbb{R}` a mapping,
# then it is possible to create the push-forward distribution :math:`\mathcal{L}` defined by
#
# .. math::
# \mathcal{L} = f(\mathcal{L}_0)
#
```

We create a 1D normal distribution

```
antecedent = ot.Normal()
```

and a 1D transform :

```
f = ot.SymbolicFunction(["x"], ["sin(x)+cos(x)"])
```

We then create the composite distribution

```
distribution = ot.CompositeDistribution(f, antecedent)
graph = distribution.drawPDF()
view = viewer.View(graph)
```

We can also build a distribution with the simplified construction

```
distribution = antecedent.exp()
graph = distribution.drawPDF()
view = viewer.View(graph)
```

and by using chained operators:

```
distribution = antecedent.abs().sqrt()
graph = distribution.drawPDF()
view = viewer.View(graph)
```

Display all figures

```
plt.show()
```