Create a random mixture

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

Create a mixture of distributions

We define an affine combination of input random variables.

Y = 2 + 5 X_1 + X_2

where:

  • X_1 \sim \mathcal{E}(\lambda=1.5)

  • X_2 \sim \mathcal{N}(\mu=4, \sigma=1)

This notion is different from the Mixture where the combination is made on the probability density functions and not on the univariate random variable.

We create the distributions associated to the input random variables :

X1 = ot.Exponential(1.5)
X2 = ot.Normal(4.0, 1.0)

We define an offset a0 :

a0 = 2.0

We create the weights :

weight = [5.0, 1.0]

We create the affine combination Y :

distribution = ot.RandomMixture([X1, X2], weight, a0)
print(distribution)

Out:

RandomMixture(Normal(mu = 6, sigma = 1) + Exponential(lambda = 0.3, gamma = 0))

We get its mean :

mean = distribution.getMean()[0]
print("Mean : %.3f"%mean)

Out:

Mean : 9.333

its variance :

variance = distribution.getCovariance()[0, 0]
print("Variance : %.3f"%variance)

Out:

Variance : 12.111

the 90% quantile :

quantile = distribution.computeQuantile(0.9)[0]
print("0.9-quantile : %.3f"%quantile)

Out:

0.9-quantile : 13.825

We can get the probability of the Y random variable to exceed 10.0 :

prb = distribution.computeSurvivalFunction(10.0)
print("Probability : %.3f"%prb)

Out:

Probability : 0.315

We draw its PDF :

graph = distribution.drawPDF()
view = viewer.View(graph)
plot create random mixture

We draw its CDF :

graph = distribution.drawCDF()
view = viewer.View(graph)
plot create random mixture

Create a discrete mixture

In this paragraph we build the distribution of the value of the sum of 20 dice rolls.

Y = \sum_{i=1}^{20} X_i

where X_i \sim U(1,2,3,4,5,6)

We create the distribution associated to the dice roll :

X = ot.UserDefined([[i] for i in range(1,7)])

Let’s roll the dice a few times !

sample = X.getSample(10)
print(sample)

Out:

    [ v0 ]
0 : [ 2  ]
1 : [ 4  ]
2 : [ 6  ]
3 : [ 6  ]
4 : [ 2  ]
5 : [ 6  ]
6 : [ 5  ]
7 : [ 2  ]
8 : [ 1  ]
9 : [ 5  ]
N = 20

We create a collection of identically distributed Xi :

coll = [X] * N

We create the weights and an affine combination :

weight = [1.0] * N
distribution = ot.RandomMixture(coll, weight)

We compute the probability to exceed a sum of 100 after 20 dice rolls :

print("Probability : %.3g"%distribution.computeComplementaryCDF(100) )

Out:

Probability : 1.58e-05

We draw its PDF :

graph = distribution.drawPDF()
view = viewer.View(graph)
X0 PDF

and its CDF :

graph = distribution.drawCDF()
view = viewer.View(graph)
X0 CDF

Display all figures

plt.show()

Total running time of the script: ( 0 minutes 0.321 seconds)

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