1.1 Cumulative Distributions
Very often it is desired to know the probability of finding x between certain limits, e.g, P(x1 x x2). This is given by the cumulative or integral distribution
where we have assumed P(x) to be continuous. If P(x) is
discrete, the integral is replaced by a sum,
By convention, also, the probability distribution is normalized to 1,
i.e.,
if x is continuous or
if x is discrete. This simply says that the probability of observing
one of the possible outcomes in a given trial is defined as 1. It
follows then that P(xi) or P(x)dx cannot be greater than 1 or less
than 0.