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]:

    

# Exactness in Lieu of CLT


    

In [3]:

    

X=ExponentialRV(1)


    

In [4]:

    

n=3


    

In [5]:

    

Y=ConvolutionIID(X,n)


    

In [6]:

    

Exact=Transform(Y,[[x/n],[0,oo]])


    

In [8]:

    

Exact.display()


    

    




continuous pdf with support [0, oo]:






    
Out[8]:





$$\begin{bmatrix}\frac{27 x^{2}}{2} e^{- 3 x}\end{bmatrix}$$

In [10]:

    

Exact.variate(s=.90)


    

    
Out[10]:





$$\begin{bmatrix}1.77410677927807\end{bmatrix}$$

In [11]:

    

CLT=NormalRV(1,1/sqrt(n))


    

In [13]:

    

CLT.variate(s=.90)


    

    
Out[13]:





$$\begin{bmatrix}1.73990414134756\end{bmatrix}$$

In [15]:

    

X=ExponentialRV(1)
sample_size=range(1,31)
percentiles=[.90,.95,.99]
header=['n','Percentile','Difference']
table=[]
for n in sample_size:
    Y=ConvolutionIID(X,n)
    Exact=Transform(Y,[[x/n],[0,oo]])
    CLT=NormalRV(1,1/sqrt(n))
    for i in percentiles:
        exact_value=Exact.variate(s=i)[0]
        clt_value=CLT.variate(s=i)[0]
        difference=exact_value-clt_value
        table.append([n,i,float(difference)])


    

In [17]:

    

import pandas as pd


    

In [18]:

    

exact_minus_clt=pd.DataFrame(table,columns=header)


    

In [22]:

    

difference_by_pct=exact_minus_clt.pivot_table('Difference',rows='n',cols='Percentile',aggfunc='sum')


    

In [24]:

    

difference_by_pct[:10]


    

    
Out[24]:






  

    

      
Percentile
      
0.9
      
0.95
      
0.99
    
    

      
n
      

      

      

    
  
  

    

      
1 
      
 0.021034
      
 0.350879
      
 1.278822
    
    

      
2 
      
 0.038666
      
 0.208845
      
 0.674200
    
    

      
3 
      
 0.034203
      
 0.148941
      
 0.458865
    
    

      
4 
      
 0.029420
      
 0.115987
      
 0.348105
    
    

      
5 
      
 0.025591
      
 0.095103
      
 0.280551
    
    

      
6 
      
 0.022588
      
 0.080664
      
 0.235020
    
    

      
7 
      
 0.020201
      
 0.070075
      
 0.202240
    
    

      
8 
      
 0.018267
      
 0.061971
      
 0.177507
    
    

      
9 
      
 0.016673
      
 0.055565
      
 0.158179
    
    

      
10
      
 0.015337
      
 0.050373
      
 0.142656
    
  

10 rows × 3 columns

In [27]:

    

difference_by_pct.plot(title='Convergence of CLT by Sample Size and Percentile')


    

    
Out[27]:





<matplotlib.axes.AxesSubplot at 0xf41f890>

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