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1.1 Cumulative Distributions

Very often it is desired to know the probability of finding x between certain limits, e.g, P(x1 leq x leq x2). This is given by the cumulative or integral distribution

Equation 1 (1)

where we have assumed P(x) to be continuous. If P(x) is discrete, the integral is replaced by a sum,

Equation 2 (2)

By convention, also, the probability distribution is normalized to 1, i.e.,

Equation 3 (3)

if x is continuous or

Equation 4 (4)

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 integ P(x)dx cannot be greater than 1 or less than 0.