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import numpy as np
import matplotlib.pyplot as pl

data = np.loadtxt('data/sizedistribution.dat');
vals = data[:,1]
num = data[:,0]

pl.plot(num,vals)

Isn't it clear that the plot is not like "random variable" ?

Exercise: Question 2, Example 3

pl.hist(vals,21)
  • 0: [565. 255. 81. 39. 21. 13. 10. 3. 2. 3. 2. 3. 1. 0. 0. 0. 0. 1. 0. 0. 1.]
  • 1: [ 1.2073 23.72266667 46.23803333 68.7534 91.26876667 113.78413333 136.2995 158.81486667 181.33023333 203.8456 226.36096667 248.87633333 271.3917 293.90706667 316.42243333 338.9378 361.45316667 383.96853333 406.4839 428.99926667 451.51463333 474.03 ]
  • 2: 
pl.hist(np.log(vals),21)
  • 0: [ 1. 4. 20. 23. 35. 48. 67. 97. 104. 107. 113. 119. 95. 57. 45. 26. 21. 6. 9. 1. 2.]
  • 1: [0.18838646 0.47280952 0.75723257 1.04165563 1.32607868 1.61050174 1.89492479 2.17934784 2.4637709 2.74819395 3.03261701 3.31704006 3.60146312 3.88588617 4.17030923 4.45473228 4.73915534 5.02357839 5.30800145 5.5924245 5.87684756 6.16127061]
  • 2: 
newvals = np.log(vals)

pl.plot(num,newvals)

print("Wrong:")
print ("Average =  %.3f" %np.mean(vals))
print ("Standard Deviation = %.3f" %np.std(vals))

print ("Correct:")
print ("Average =  %.3f" % np.exp(np.mean(newvals)))
print ("Standard Deviation = %.3f" % np.exp(np.std(newvals,ddof=1)))

Wrong:
Average =  32.048
Standard Deviation = 39.773
Correct:
Average =  19.980
Standard Deviation = 2.637

from scipy.stats import lognorm, norm
param = norm.fit(newvals)

x = np.linspace(np.min(newvals),np.max(newvals),100)
# fitted distribution
pdf_fitted = norm.pdf(x,loc=param[0],scale=param[1])

pl.figure()
pl.plot(x,pdf_fitted,'r-')
pl.hist(newvals,density=1,alpha=.3)
  • 0: [0.0100454 0.08036319 0.15737791 0.30471041 0.39679323 0.38842207 0.20927913 0.08873435 0.03348466 0.0050227 ]
  • 1: [0.18838646 0.78567488 1.38296329 1.98025171 2.57754012 3.17482854 3.77211695 4.36940537 4.96669378 5.5639822 6.16127061]
  • 2: 
print(param) # log normal
print(np.exp(param))

(np.float64(2.9947081784485516), np.float64(0.9690344007390227))
[19.97952858  2.63539849]