density distribution vs. cumulative density function
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Cumulative density is cumulative properties.

 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) 
 
  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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standard normal and cumulative normal distributions
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The standard normal distribution is defined by the following equation: 
                       1                                     
           f(x) = --------- * exp(-0.5 *x^2)               
                   sqrt(2*pi)                                

    where f(x) is the density function, x is the input
       pi is 3.1415926

 Manually, we could calculate a few values. 
    x=0    -> f(x) = 1/sqrt(2*3.1415925) ->  0.3989423

    x=1    -> f(1) = 1/sqrt(2*3.1415926)*exp(-0.5)          ->   0.2419707  
    x=-1    -> f(-1) = 1/sqrt(2*3.1415926)*exp(-0.5*(-1)^2) ->   0.2419707

 The normal distribution is a probability distribution that associates 
    the normal random variable X with a cumulative probability . 
    The normal distribution is defined by the following equation:
                           1                                     
           f(x) = ---------------- * exp(-0.5 *[(x-mean)/(sigma))]^2)               
                 sqrt(2*pi*sigma^2)                                

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