Expected Value
The Expected Value of the Normal Curve in Red; goes through the area of the Standard Deviation and follows the triangular relationship that exist.
The Red Curve E(x)
Probabilities are coming from the Standard Deviation with the total area of the Standard Deviation equals the Constant Value .3988 which is equal to the Normal Constant- 1/(2*pi)^.5
I wonder if ?? that %40 of the area within the range of (0,1) have %40 of the Deviations that calculate
the Normal Probabilities while the other %10 occur in the tail -- thinking I can separate the data by saying
%40 of the data's deviations are happening within 1 Standard Deviation while %10 of the data is in the tail; At Sigma = 100 or Z = .01
while Sigma .2222 or Z = 4.5 has % 11 of the data within 1 Standard Deviation while the other %39 percent lie beyond .851 which is the Limit of integration of RV Sigma. ????
E(x) shows an Expected value within 1 Std. I guess the Expected Value moves to the tail as the Deviation increases while a decrease in Deviation puts the Expected values within 1 STD.
But,, the Deviation of the Standard is arbitrary; but which ever point I chose to integrate, the formula still would have a ratio of Integrated area / total area which makes me wonder where the data
really resides????
I just notice that the value of Sigma also seems to be the peak or maximum, mean location number, for the Expected Value in Red> Look At Sigma = 2 , 3 , 4 , 5 , 6 and it seems, the red curves maximum are at those sigma values and continue in
that fashion!!