dsfs 06 probability

probability

1. dependence and independence

   1. e and f are independent

       P(E,F) = P(E)P(F)
      

2. conditional probability

   1. e and f are not independent

       P(E|F) = P(E,F)/P(F)
      
       # rewrite as this
       P(E,F) = P(E|F)P(F)
      
       # when e and f are independent
       P(E|F) = P(E)
      

   2. b both children are girls g the older child is a girl

       P(B|G) = P(B,G)/P(G) = P(B)/P(G) = 1/2
      

   3. b both children are gilrs l at the least one of them is a girl

       P(B|L) = P(B,L)/P(L) = P(B)/P(L) = 1/3
      

   4. check

       def random_kid():
           return random.choice(['boy', 'girl'])
      
       both_girls = 0
       older_girl = 0
       either_girl = 0
       random.seed(0)
       for _ in range(10000):
           younger = random_kid()
           older = random_kid()
           if older == 'girl':
               older_girl += 1
           if older == 'girl' and younger == 'girl':
               both_girls += 1
           if older == 'girl' or younger == 'girl':
               either_girl += 1
      
       print 'p(both | older)', both_girls / older_girl    # 0.514 ~ 1/2
       print 'p(both | either)', both_girls / either_girl  # 0.342 ~ 1/3
      

3. bayes’s theorem

        P(E|F) = P(E,F)/P(F) = P(F|E)P(E)/P(F)
   
        # given
        P(F) = P(F,E) + P(F,¬E)
   
        # so
        P(E|F) = P(F|E)P(E)/[P(F|E)P(E)+P(F|¬E)P(¬E)]
   

continuous distribution

1. probability density function pdf

   1. density function for the uniform distribution

       def uniform_pdf(f):
           return 1 if x >= 0 and x < 1 else 0
      

2. cumulative distribution function cdf

        def uniform_cdf(x):
            """returns the probability that a uniform random variable is <= x"""
   
            # uniform random is never less than 0
            if x < 0 return 0
            # e.g. P(X <= 0.4) = 0.4
            elif x < 1 return x
            #uniform random is aways less than 1
            else: return 1
   

normal distribution