Distributions and inverse distributions

### 19.4.1 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) dx=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(X ≤ x)=h.