Outline
        h Lecture 1
           hDescriptive Statistics
           hProbability & Likelihood
        h Lecture 2
           hThe Frequentist Approach 
           hThe Bayesian Approach
        h Lecture 3
           hAnalysis Example
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      Descriptive Statistics
                Descriptive Statistics – 1
      Definition: A statistic is any function of the data X.
      Given a sample X = x , x , … x , it is often of interest 
                         1 2     N
      to compute statistics such as 
                                  1 N
      the sample average      x     x
                                  N      i
                                     i1
                                    1 N
      and the sample variance  S2    (x  x)2
                                    N       i
                                       i1
      In any analysis, it is good practice to study ensemble 
         averages, denoted by < … >, of relevant statistics
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                                      Descriptive Statistics – 2
              Ensemble Average                                                 x
              Mean                                                              
              Error                                                              x 
              Bias                                                              bx
              Variance                                                                                                                2
                                                                               V =<(x−) >
                                                                               
              Mean Square Error                                                                                                      2
                                                                               MSE=<(x−μ) >
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                Descriptive Statistics – 3
                      MSE(x )2                    Exercise 1:
                                       2               Show this
                             Vb
       The MSE is the most widely used measure of closeness of an
       ensemble of statistics {x} to the true value μ 
       The root mean square (RMS) is
                        RMS MSE
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