Auction Theory, 2nd Edition

Appendix | A

Continuous

Distributions

Given a random variable X, which takes on values in [0,ω], we define its

cumulative distribution function F : [0, ω]→[0, 1] by

1

F(x) =Prob[X ≤x] (A.1)

the probability that X takes on a value not exceeding x. By definition, the func-

tion F is nondecreasing and satisfies F

(

0

)

=0 and F

(

ω

)

=1 (if ω =∞, then

lim

x→∞

F(x) =1). In this book we always suppose that F is increasing and

continuously differentiable.

The derivative of F is called the associated probability density function and

is usually denoted by the corresponding lowercase letter f ≡F



. By assumption,

f is continuous and we will suppose, in addition, that for all