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In [1]:



    


import pandas as pd
import numpy as np
from math import *

from ggplot import *
%matplotlib inline



    











In [4]:



    


x = np.random.uniform(size=100000)
df = pd.DataFrame({'x':x})



    











In [5]:



    


df



    


















    
Out[5]:










  
    

      
      x
    

  
  
    

      0
      0.072823
    

    

      1
      0.333318
    

    

      2
      0.589766
    

    

      3
      0.725427
    

    

      4
      0.254256
    

    

      5
      0.994160
    

    

      6
      0.356303
    

    

      7
      0.968033
    

    

      8
      0.526983
    

    

      9
      0.232518
    

    

      10
      0.527262
    

    

      11
      0.683994
    

    

      12
      0.069104
    

    

      13
      0.442857
    

    

      14
      0.972388
    

    

      15
      0.369894
    

    

      16
      0.624445
    

    

      17
      0.072818
    

    

      18
      0.508043
    

    

      19
      0.267760
    

    

      20
      0.634653
    

    

      21
      0.382272
    

    

      22
      0.185888
    

    

      23
      0.670246
    

    

      24
      0.447122
    

    

      25
      0.930639
    

    

      26
      0.037773
    

    

      27
      0.762964
    

    

      28
      0.338128
    

    

      29
      0.308757
    

    

      ...
      ...
    

    

      99970
      0.968960
    

    

      99971
      0.976037
    

    

      99972
      0.599949
    

    

      99973
      0.062972
    

    

      99974
      0.222659
    

    

      99975
      0.425420
    

    

      99976
      0.364649
    

    

      99977
      0.853154
    

    

      99978
      0.868834
    

    

      99979
      0.302208
    

    

      99980
      0.065341
    

    

      99981
      0.176708
    

    

      99982
      0.400384
    

    

      99983
      0.133862
    

    

      99984
      0.561546
    

    

      99985
      0.907003
    

    

      99986
      0.873324
    

    

      99987
      0.553295
    

    

      99988
      0.743294
    

    

      99989
      0.208195
    

    

      99990
      0.467326
    

    

      99991
      0.074584
    

    

      99992
      0.230796
    

    

      99993
      0.313990
    

    

      99994
      0.585190
    

    

      99995
      0.411894
    

    

      99996
      0.795058
    

    

      99997
      0.907080
    

    

      99998
      0.568661
    

    

      99999
      0.434893
    

  



100000 rows × 1 columns



















In [6]:



    


ggplot(aes(x='x'), data=df) + geom_histogram()



    


















    






stat_bin: binwidth defaulted to range/30.
    Use 'binwidth = x' to adjust this.










    

















    
Out[6]:








<ggplot: (281526977)>

















In [9]:



    


%%latex
x = y



    


















    







x = y

Inversion sampling example

First find the normalizing constant: $$ \begin{align*} f_X(X=x) &&= cx^2, 0 \leq x \leq 2 \\ 1 &&= c\int_0^2 x^2 dx \\ &&= c[\frac{1}{3}x^3 + d]_0^2 \\ &&= c[\frac{8}{3} + d - d] \\ &&= c[\frac{8}{3}] \\ f_X(X=x) &&= \frac{3}{8}x^2, 0 \leq x \leq 2 \end{align*} $$

Next find the cumulative distribution function:

  • $F_X(X=x) = \int_0^x \frac{3}{8}x^2dx$
  • $=\frac{3}{8}[\frac{1}{3}x^3 + d]_0^x$
  • $=\frac{3}{8}[\frac{1}{3}x^3 + d - d]$
  • $=\frac{1}{8}x^3$

We can randomly generate values from a standard uniform distribution and set equal to the CDF. Solve for $x$. Plug the randomly generated values into the equation and plot the histogram or density of $x$ to get the shape of the distribution:

  • $u = \frac{1}{8}x^3$
  • $x^3 = 8u$
  • $x = 2u^{\frac{1}{3}}$




In [13]:



    


%%latex

\begin{align*}
    f_X(X=x) &= cx^2, 0 \leq x \leq 2 \\
    1 &= c\int_0^2 x^2 dx \\
    &= c[\frac{1}{3}x^3 + d]_0^2 \\
    &= c[\frac{8}{3} + d - d] \\
    &= c[\frac{8}{3}] \\
    f_X(X=x) &= \frac{3}{8}x^2, 0 \leq x \leq 2
\end{align*}



    


















    








\begin{align*}
    f_X(X=x) &= cx^2, 0 \leq x \leq 2 \\
    1 &= c\int_0^2 x^2 dx \\
    &= c[\frac{1}{3}x^3 + d]_0^2 \\
    &= c[\frac{8}{3} + d - d] \\
    &= c[\frac{8}{3}] \\
    f_X(X=x) &= \frac{3}{8}x^2, 0 \leq x \leq 2
\end{align*}
















In [41]:



    


u = np.random.uniform(size=100000)
x = 2 * u**.3333

df = pd.DataFrame({'x':x})
print df.describe()
ggplot(aes(x='x'), data=df) + geom_histogram()



    


















    






                   x
count  100000.000000
mean        1.499664
std         0.387398
min         0.021286
25%         1.260849
50%         1.587595
75%         1.816844
max         1.999999










    

















    
Out[41]:








<ggplot: (286026045)>

Joint Distribution

Find the normalizing constant:

  • $f_{X,Y}(X=x,Y=y) = c(2x + y), 0 \leq x \leq 2, 0 \leq y \leq 2$
  • $1 = \int_0^2 \int_0^2 c(2x + y) dy dx$
  • $ = c\int_0^2 [2xy + \frac{1}{2}y^2 + d]_0^2 dx$
  • $ = c\int_0^2 [4x + \frac{1}{2}4 + d - d] dx$
  • $ = c\int_0^2 (4x + 2) dx$
  • $ = c[2x^2 + 2x + d]_0^2$
  • $ = c[2(4) + 2(2) + d - d]$
  • $ = 12c$
  • $c = \frac{1}{12}$
  • $f_{X,Y}(X=x,Y=y) = \frac{1}{12}(2x + y), 0 \leq x \leq 2, 0 \leq y \leq 2$




In [5]:



    


x = np.random.uniform(size=10000)
y = np.random.uniform(size=10000)

Find the marginal distribution:

  • $f_{X,Y}(X=x,Y=y) = \frac{1}{12}(2x + y), 0 \leq x \leq 2, 0 \leq y \leq 2$
  • $f_X(X=x) = \int_0^2 \frac{1}{12}(2x + y) dy$
  • $ = \frac{1}{12}[2xy + \frac{1}{2}y^2 + d]_0^2$
  • $ = \frac{1}{12}[4x + 2 + d - d]$
  • $ = \frac{4x + 2}{12}$
  • $ = \frac{2x + 1}{6}$

Inversion sampling example:

  • $F_X(X=x) = \int_0^x \dfrac{2x+1}{6}dx$
  • $= \frac{1}{6}[x^2 + x + d]_0^x$
  • $= \frac{x(x + 1)}{6}$
  • $u = \frac{x^2 + x}{6}$
  • $0 = x^2 + x - 6u$
  • $x = \frac{-1 \pm \sqrt{1 + 4 \times 6u}}{2}$




In [8]:



    


u = np.random.uniform(size=100000)
x = (-1 + (1 + 24*u)**.5) / 2
df = pd.DataFrame({'x':x})
ggplot(aes(x='x'), data=df) + geom_histogram()



    


















    

















    
Out[8]:








<ggplot: (279378645)>

















In [ ]:

Content source: jtwalsh0/methods

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