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9.4.1  Distributions and inverse distributions

Let p(x) be a probability density function, so p(x) ≥ 0 for all x, and for a discrete density function,

 
x∈ ℤ
 p(x) = 1

while for a continuous density function,



−∞
 p(x) = 1

The corresponding cumulative distribution function

P(x)= Prob(X ≤ x)

is the probability that a randomly (according to the probability being considered) chosen value is less than or equal to x. This can be used to find the probability that a randomly chosen value is between two numbers:

Prob(x < X ≤ y) = P(y) − P(x)

Given a value h between 0 and 1, the inverse distribution function for a distribution takes h to the value of x for which Prob(Xx) = h.


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