python11():
  Exercises for random numbers, simulation, testing.

  Initialize random number generator with seed =  123456789
  Five random integers 0 <= i < 100:
[90  2 58 90 34 88 29 62 12 79]

  Initialize random number generator with seed =  987654321
  Five random integers 0 <= i < 100:
[81 28 77 17 92 19 25 36 83 14]

  Initialize random number generator with seed =  123456789
  Five random integers 0 <= i < 100:
[90  2 58 90 34 88 29 62 12 79]

  r = rng.integers ( low = 0, high = 100 )
19
  r = rng.integers ( low = 0, high = 100, size = 1 )
[82]
  r = rng.integers ( low = 0, high = 100, size = 3 )
[41 84  6]
  r = rng.integers ( low = 0, high = 100, size = ( 3, 2 ) )
[[47 24]
 [95 82]
 [94 65]]

  Return random floats in [0,1]

  r = rng.random ( )
0.08364013148965865
  rng.random ( 1 )
[0.26984426]
  rng.random ( 3 )
[0.05379984 0.56661849 0.01891774]
  rng.random ( ( 3, 2 ) )
[[0.20537362 0.24935483]
 [0.89877802 0.10779729]
 [0.74867811 0.71470362]]

  statistics for random():

  x = rng._random ( 10000 )
  x[0:5] =  [0.74077908 0.94511748 0.45946087 0.23163497 0.46497377]
  mean ( x ) =  0.5024079988122446  Distribution mean = 0
  std ( x ) =  0.2887575177771944  Distribution std =  0.2886751345948129
  min ( x ) =  2.084883961028794e-05  Distribution min = 0.0
  min ( x ) =  0.9999101448053969  Distribution max = 1.0

  Return random floats in [ 3.141592653589793 , 10.0 ]

  r = a + ( b - a ) * rng.random ( )
9.543135325260273
  r = a + ( b - a ) * rng.random ( 1 )
[5.31041253]
  r = a + ( b - a ) * rng.random ( 3 )
[6.87005209 3.83932539 9.22829008]
  r = a + ( b - a ) * rng.random ( ( 3, 2 ) )
[[6.98204939 6.18873156]
 [5.82288157 7.94788268]
 [7.74022727 3.41641799]]

  r = rng.standard_normal ( )
0.6227622538674881
  rng.standard_normal ( 1 )
[-1.25521346]
  rng.standard_normal ( 3 )
[-0.36810849  0.1818789   1.5069514 ]
  rng.standard_normal ( ( 3, 2 ) )
[[0.62958443 0.1031141 ]
 [1.07140017 1.60669513]
 [1.35884452 1.48410916]]

  standard_normal:

  x = rng._standard_normal ( 10000 )
  x[0:5] =  [ 0.21711346 -0.25249231 -0.38976507  0.18114662 -1.50870976]
  mean ( x ) =  -0.005161080790342771  Distribution mean = 0.0
  std ( x ) =  1.0094860025727166  Distribution std = 1.0
  min ( x ) =  -4.234554911424464  Distribution min = -inf
  max ( x ) =  4.055359783888028  Distribution max = +inf

  normal:

  x = rng.normal (  10.0 , 2.0 , 1000 )
  x[0:5] =  [10.3329046   9.75213545 10.78131338 10.44510029  7.12891626]
  mean ( x ) =  10.000603438014656  Distribution mean =  10.0
  std ( x ) =  2.0154419861609827  Distribution std =  2.0
  min ( x ) =  1.5983534388648746  Distribution min = -inf
  max ( x ) =  17.15938314341716  Distribution max = +inf

  Randomly permute a vector.

  a = np.arange ( 10 )
[0 1 2 3 4 5 6 7 8 9]
  b = np.permutation ( a )
[9 8 1 2 5 4 7 3 6 0]

  Randomly permute a matrix.

  c:
[[ 0  1  2  3  4]
 [10 11 12 13 14]
 [20 21 22 23 24]]
  d = rng.permutation ( c )
[[20 21 22 23 24]
 [10 11 12 13 14]
 [ 0  1  2  3  4]]
  e = np.transpose ( d )
[[20 10  0]
 [21 11  1]
 [22 12  2]
 [23 13  3]
 [24 14  4]]
  f = rng.permutation ( e )
[[21 11  1]
 [22 12  2]
 [23 13  3]
 [24 14  4]
 [20 10  0]]
  g = rng.transpose ( e )
[[21 22 23 24 20]
 [11 12 13 14 10]
 [ 1  2  3  4  0]]
  defg = np.transpose ( rng.permutation ( np.transpose ( rng.permutation ( c ) ) ) )
[[22 20 21 23 24]
 [12 10 11 13 14]
 [ 2  0  1  3  4]]

  Randomly select 3 items from 12, using permutation.

<class 'range'>
<class 'numpy.ndarray'>
['Jun' 'Feb' 'Apr']

  Randomly select 5 cards from 52, no repeats

  cards = np.arange ( 52 )
[ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
 48 49 50 51]
  b = rng.permutation ( cards )
[33  4 41 40 37 29 44 17  3 15 35 32 51 31 34 50 23  7 46 11  1  8 16 25
 45  5 21 18 48 28 49 24 30 19 27 14  9 47 42 38  2 26 43 20 10 13 39 12
  0 36  6 22]
  c = b[0:5]
[33  4 41 40 37]

  Randomly select 5 cards from 52, can have repeats

  cards = np.arange ( 52 )
[ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
 48 49 50 51]
  b = rng.choice ( cards, size = 5 )
[47 39  2 37 14]

  Randomly select W/L/T values for a game season

  winlosstie = [ 'win', 'loss', 'tie' ]
['win', 'loss', 'tie']
  record = rng.choice ( winlosstie, size = 10 )
['loss' 'loss' 'loss' 'win' 'win' 'win' 'tie' 'tie' 'loss' 'loss']
  record = rng.choice ( winlosstie, size = 10 )
['win' 'tie' 'win' 'tie' 'win' 'tie' 'tie' 'loss' 'loss' 'tie']

  Generate a binary matrix.

  binary = [ 0, 1 ]
[0, 1]
  binary_matrix = rng.choice ( binary, size = ( 4, 5 ) )
[[0 0 0 1 1]
 [1 0 0 1 1]
 [1 1 0 0 0]
 [1 0 1 0 0]]

  Compute estimate Q for integral(a<=x<=b) f(x) dx

  N    Q    Error

10 0.8392960364153677 0.07626788466784307
100 0.7017843825644043 0.061243769183120245
1000 0.7652918738910668 0.0022637221435422017

  Compute estimate Q for area of ellipse

  N    Q    Error

10 8.0 1.7168146928204138
100 6.16 0.12318530717958609
1000 6.264 0.019185307179585998

  Estimate average distance between two points in unit square.

  N    Estimate  Error

10 0.5670205557895164 0.04561512262479572
100 0.5063219136715147 0.015083519493205966
1000 0.5148028843337755 0.006602548830945132
10000 0.5231199707619855 0.0017145375972648091

  ||A||2 <= ||A||F always?

  Statement was true  1000  times out of  1000