stochastic programsprobabilityand statistics notice that while the means in both plots are about the samethe standard deviations are quite different the spread of outcomes is much tighter when we flip the coin times per trial than when we flip the coin times per trial to make this clearwe have used pylab xlim to force the bounds of the -axis in the second plot to match those in the first plotrather than letting pylab choose the bounds we have also used pylab xlim and pylab ylim to choose set of coordinates for displaying text box with the mean and standard deviation normal distributions and confidence levels the distribution of results in each of these plots is close to what is called normal distribution technically speakinga normal distribution is defined by the formula *(!!!)!where is the meanthe standard deviationand euler' number (roughly if you don' feel like studying this equationthat' fine just remember that normal distributions peak at the meanfall off symmetrically above and below the meanand asymptotically approach they have the nice mathematical property of being completely specified by two parametersthe mean and the standard deviation (the only two parameters in the equationknowing these is equivalent to knowing the entire distribution the shape of the normal distribution resembles (in the eyes of somethat of bellso it sometimes is referred to as bell curve as we can see by zooming in on the center of the plot for flips/trialthe distribution is not perfectly symmetricaland therefore not quite normal howeveras we increase the number of trialsthe distribution will converge towards normal normal distributions are frequently used in constructing probabilistic models for three reasons they have nice mathematical properties many naturally occurring distributions are indeed close to normaland they can be used to produce confidence intervals instead of estimating an unknown parameter by single value ( the mean of set of trials) confidence interval provides range that is likely to contain the unknown value and degree of confidence that the unknown value lies within that range for examplea political poll might indicate that candidate is likely to get of the vote +- ( the confidence interval is of size with confidence level of what this means is that the pollster believes that of the time the candidate will receive between and of the vote together the confidence interval and the confidence level indicate the reliability of the