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Last modified: January 26, 2018

Expected Value
--------------

The expected value is also known as the average, mean 
or first moment.  It is defined as:

    N
E = ΣxiP(xi)
    i
Where x is a random variable and P(x) is the probability 
of occurence.

Example:  Dice.

1 + 2 + 3 + 4 + 5 + 6 = 21  Average = Σx/Nn= 21/6 = 3.5

1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5

Variance
--------
            _            
s2 = E[(x - x)2] 
            _
    ≡ Σ(x - x)2/N
              _    _
    = E[x2 - 2xx + x2]
               _       _      
    = E(x2) - 2xE(x) + x2E(1)
                                              _
    = E(x2) - 2E(x)E(x) + E(x)E(x)E(1)  since x = E(x)

    = E(x2) - E(x)E(x) 

    = Σx2/N - (Σx/N)2

    = NΣx2 - (Σx)2
      ------------
          N2

This is the computational form for the variance.

Skewness
--------
             _
      Σ(xi - x)3/N           _
g1 =  ------------ = E[(xi - x)3/s3]
          s3

Kurtosis
--------
             _
      Σ(xi - x)4/N           _
g2 =  ------------ = E[(xi - x)4/s4]
          s4

 

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