**1.1 Cumulative Distributions**

Very often it is desired to know the probability of finding x between
certain limits, e.g, *P*(*x*_{1} *x* *x*_{2}). 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(x _{i})* or