Concepts of distributions and cumulative distributions
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Properties of a probability:  1) prob(i) >=0                                   
                              2) summation of all is 1

  Case #1: Toss a coin, what is the probability of a head?  
              head is 50%, i.e., 50% vs. 50%                   

  Case #2: 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) 
 
  Case #3: 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   
 
  Case #4: Standard normal distribution                    
           The density function is            1                                     
                                  f(x) = --------- * exp(-0.5 *x^2)               
                                         sqrt(2*pi)                                
    
    When x=0, f(0) = 1/sqrt(2*3.1425926)=0.3988788          
     Using Excel =normdist(0,0,1,FALSE) = 0.398942280401433       
       
  Case #5: Cumulative standard normal distribution. Let's choose a value of 1                      
           meaning: summation of all small strips from negative infinity up to x=1                      
           Using Excel =normdist(1,0,1,TRUE)  -> 0.841344746068543        

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