45-733 PROBABILITY AND STATISTICS I Notes #4B


February 2000



  1. Marginal Distributions
    Definition: Continuous Marginal Distributions
    f1(x) = ò-¥+¥ f(x,y)dy
    f2(y) = ò-¥+¥ f(x,y)dx
    Definition: Discrete Marginal Distributions
    f1(xi) = åj=1,m f(xi, yj)
    f2(yj) = åi=1,n f(xi, yj)

  2. Example:
    
                               æ (1/8)x2  0 < x < 2
                               ç          0 < y < 3
                      f(x,y) = ç
                               ç
                               è   0 otherwise
    

    1. Find f1(x) and f2(y)
      f1(x) = ò03 (1/8)(x2)dy = (1/8)(x2)y|03 = (3/8)x2. That is:
      
                                 æ (3/8)x2  0 < x < 2
                         f1(x) = ç
                                 è   0 otherwise
      


      f2(y) = ò02 (1/8)(x2)dx = [(x3)/24]|02 = 8/24 = 1/3. That is:
      
                                 æ 1/3  0 < y < 3
                         f2(y) = ç
                                 è  0 otherwise
      

      Note that f2(y) is a continuous uniform distribution.