Review: variance vs. standard deviation 
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For a given set of returns, we have the following formulae 
    to estimate variance and standard deviation. 

 Assume that our a set of n returns: R1, R2, ... Rn
                     R1+ R2 + ... + Rn
    Step 1: mean= ------------------------
                            n

             (R1-mean)^2 + (R2-mean)^2 + ... + (Rn-mean)^2
    var = -------------------------------------------
                                  n-1
    std = sqrt(var)

  Note : usually, when the IBM's volatility is 0.2 means that its 
         annualized standard deviation of returns is 0.2.

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Review: density vs. cumulative distribution  
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Cumulative density has a cumulative property.
Discrete cases
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 Case #1: Toss a dice with values from 1 to 6           
             Each number has 1/6 probability                
             What is the prob. to get a number less than 4?  
                      1/6 + 1/6 + 1/6 = 3/6 =1/2                
             1/6 is probability of each number, while           
             1/2 is the cumulative probability (distribution) 
Continuous cases
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  Case #2: uniform distribution: assume density function is 1/a and choose a=5    
            from 1 to 5, the density is 1/5. total sum is 5*1/5 =1                               
            what does it mean that the density function is 1/5 (0.2)?                              
            Take a value between 0 and 5, such as x=1.5 and choose a small number 
            about called deltaX. The probability that a random number falls into  
               this small strip = f(x) * deltaX= 0.2*detlaX   
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Review: left tail 
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Since the negative is on the left-hand side, 

     we use the left tail to represent loss

     or risk. 

The loss of one 1% is the area of the left tail
   of a normal distribution (1%). 
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