In [1]:

    

from applpy import *


    

    




-----------------
Welcome to ApplPy
-----------------
ApplPy Procedures

Procedure Notation

Capital letters are random variables
Lower case letters are number
Greek letters are parameters
gX indicates a function
n and r are positive integers where n>=r
Square brackets [] denote a list
Curly bracks {} denote an optional variable


RV Class Procedures
X.variate(n,x),X.verifyPDF()

Functional Form Conversion
CDF(X,{x}),CHF(X,{x}),HF(X,{x}),IDF(X,{x})
PDF(X,{x}),SF(X,{x}),BootstrapRV([data])
Convert(X,{x})

Procedures on One Random Variable
ConvolutionIID(X,n),CoefOfVar(X),ExpectedValue(X,gx)
Kurtosis(X),MaximumIID(X,n),Mean(X),MGF(X)
MinimumIID(X,n),OrderStat(X,n,r),ProductIID(X,n)
Skewness(X),Transform(X,gX),Truncate(X,[x1,x2])
Variance(X)

Procedures on Two Random Variables
Convolution(X,Y),Maximum(X,Y),Minimum(X,Y)
Mixture([p1,p2],[X,Y]),Product(X,Y)

Statistics Procedures
KSTest(X,[sample]), MOM(X,[sample],[parameters])
MLE(X,[sample],[parameters],censor)

Utilities
PlotDist(X,{[x1,x2]}),PlotDisplay([plotlist],{[x1,x2]})
PPPlot(X,[sample]),QQPlot(X,[sample])

Bayesian Procedures
CredibleSet(X,alpha), JeffreysPrior(X,low,high,param)
Posterior(X,Y,[data],param)
PosteriorPredictive(X,Y,[data],param)

Continuous Distributions
ArcSinRV(),ArcTanRV(alpha,phi),BetaRV(alpha,beta)
CauchyRV(a,alpha)ChiRV(N),ChiSquareRV(N),ErlangRV(theta,N)
ErrorRV(mu,alpha,d),ErrorIIRV(a,b,c),ExponentialRV(theta)
ExponentialPowerRV(theta,kappa),ExtremeValueRV(alpha,beta)
FRV(n1,n2),GammaRV(theta,kappa),GompertzRV(theta,kappa)
GeneralizedParetoRV(theta,delta,kappa),IDBRV(theta,delta,kappa)
InverseGaussianRV(theta,mu),InverseGammaRV(alpha,beta)
KSRV(n),LaPlaceRV(omega,theta), LogGammaRV(alpha,beta)
LogisticRV(kappa,theta),LogLogisticRV(theta,kappa)
LogNormalRV(mu,sigma),LomaxRV(kappa,theta)
MakehamRV(theta,delta,kappa),MuthRV(kappa),NormalRV(mu,sigma)
ParetoRV(theta,kappa),RayleighRV(theta),TriangularRV(a,b,c)
TRV(N),UniformRV(a,b),WeibullRV(theta,kappa)

Discrete Distributions
BenfordRV(),BinomialRV(n,p),GeometricRV(p),PoissonRV(theta)
IPython console for SymPy 0.7.5 (Python 2.7.6-64-bit) (ground types: python)

These commands were executed:
>>> from __future__ import division
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)

Documentation can be found at http://www.sympy.org






    




WARNING: Hook shutdown_hook is deprecated. Use the atexit module instead.

In [2]:

    

X=ExponentialRV(Rational(1,7))


    

In [3]:

    

Sys=MinimumIID(X,4)


    

In [4]:

    

sys_mean=Mean(Sys)


    

In [5]:

    

sys_mean


    

    
Out[5]:





$$\frac{7}{4}$$

In [15]:

    

n=1
header=['n','System Mean']
sys_data=[]
while n<15:
    SubSys1=MinimumIID(X,3)
    SubSys2=MaximumIID(X,n)
    Sys=Minimum(SubSys1,SubSys2)
    sys_mean=float(Mean(Sys))
    sys_data.append([n,sys_mean])
    n+=1


    

In [16]:

    

import pandas as pd


    

In [17]:

    

sys_lifetime=pd.DataFrame(sys_data,columns=header)


    

In [19]:

    

mean_lifetime_by_n=sys_lifetime.pivot_table('System Mean',rows='n',aggfunc='sum')


    

In [20]:

    

mean_lifetime_by_n


    

    
Out[20]:





n
1     1.750000
2     2.100000
3     2.216667
4     2.266667
5     2.291667
6     2.305556
7     2.313889
8     2.319192
9     2.322727
10    2.325175
11    2.326923
12    2.328205
13    2.329167
14    2.329902
Name: System Mean, dtype: float64

In [21]:

    

SubSys1=MinimumIID(X,3);SubSys2=MaximumIID(X,8)


    

In [22]:

    

Sys=Minimum(SubSys1,SubSys2)


    

In [24]:

    

Sys.display()


    

    




continuous cdf with support [0, oo]:






    
Out[24]:





$$\begin{bmatrix}\left(e^{\frac{x}{7}} - 1\right)^{8} e^{- \frac{11 x}{7}} + 1 - e^{- \frac{3 x}{7}}\end{bmatrix}$$

In [25]:

    

Mean(Sys)


    

    
Out[25]:





$$\frac{1148}{495}$$

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