UE Stochastic Processes - Beispiel 10

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Für zwei Wahrscheinlichkeitsvektoren \mu = (\mu_1, \mu_2, \ldots, \mu_n) \, und \nu = (\nu_1, \nu_2, \ldots, \nu_n) \, zeige man

 ||\mu - \nu||_1 = 2 [ 1 - \sum_{i=1}^n \min(\mu_i,\nu_i)] \,,

wobei ||\mu - \nu||_1 =\sum_{i=1}^n | \mu_i - \nu_i | \,.


 2 [ 1 - \sum_{i=1}^n \min(\mu_i,\nu_i)] = 
 2 - 2 \sum_{i=1}^n \min(\mu_i,\nu_i)] = 
 \sum_{i=1}^n \mu_i + \sum_{i=1}^n \nu_i - \sum_{i=1}^n \min(\mu_i,\nu_i)] = 
 

  = \sum_{i=1}^n [ \mu_i + \nu_i - 2 \min(\mu_i,\nu_i)] = 
  \sum_{i=1}^n | \mu_i - \nu_i | =
  ||\mu - \nu||_1
 

für \mu_i > \nu_i: \mu_i + \nu_i - 2 \nu_i = \mu_i - \nu_i \, 

für \mu_i < \nu_i: \mu_i + \nu_i - 2 \mu_i = \nu_i - \mu_i \, 

für \mu_i = \nu_i: \mu_i + \mu_i - 2 \mu_i = 0 \,