Springer - Read

A Random
Variables and
Probability Distributions
A.1 Distribution Functions and Expectation
A.2 Random Vectors
A.3 The Multivariate Normal Distribution
A.1 Distribution Functions and Expectation
The distribution function F of a random variable X is defined by
F(x) P [X ≤ x] (A.1.1)
for all real x. The following properties are direct consequences of (A.1.1):
1. F is nondecreasing, i.e., F(x) ≤ F(y)if x ≤ y.
2. F is right continuous, i.e., F(y) ↓ F(x)as y ↓ x.
3. F(x) → 1 and F(y) → 0asx →∞and y →−∞, respectively.
Conversely, any function that satisfies properties 1–3 is the distribution function of
some random variable.
Most of the commonly encountered distribution functions F can be expressed
either as
or
F(x)
x
−∞
f(y)dy (A.1.2)
F(x)
p(xj), (A.1.3)
j:xj ≤x