Distribution and density functions

Introduction

In the unit on Random Variables and Probability we introduce real random variables as mappings from the basic space Ω to the real line. The mapping induces a transfer of the probability mass on the basic space to subsets of the real line in such a way that the probability that X takes a value in a set M is exactly the mass assigned to that set by the transfer. To perform probability calculations, we need to describe analytically the distribution on the line. Forsimple random variables this is easy. We have at each possible value of X a point mass equal to the probability X takes that value. For more general cases, we need a more useful description than that provided by the induced probability measure P _X .

The distribution function

In the theoretical discussion on Random Variables and Probability , we note that the probability distribution induced bya random variable X is determined uniquely by a consistent assignment of mass to semi-infinite intervals of the form $(-\infty ,t]$ for each real t . This suggests that a natural description is provided by the following.

Definition

The distribution function F _X for random variable X is given by

In terms of the mass distribution on the line, this is the probability mass at or to the left of the point t . As a consequence, F _X has the following properties:

• (F1) F _X must be a nondecreasing function, for if $t>s$ there must be at least as much probability mass at or to the left of t as there is for s .
• (F2) F _X is continuous from the right, with a jump in the amount p ₀ at t ₀ iff $P(X=t_0)=p_0$ . If the point t approaches t ₀ from the left, the interval does not include the probability mass at t ₀ until t reaches that value, at which point the amount at or to the left of t increases ("jumps") by amount p ₀ ; on the other hand, if t approaches t ₀ from the right, the interval includes the mass p ₀ all the way to and including t ₀ , but drops immediately as t moves to the left of t ₀ .
• (F3) Except in very unusual cases involving random variables which may take “infinite” values, the probability mass included in $(-\infty ,t]$ must increase to one as t moves to the right; as t moves to the left, the probability mass included must decrease to zero, so that
  $$F_X(-\infty )=\underset{t\to -\infty }{lim}{F}_X\left(t\right)=0\text{and}{F}_X\left(\infty \right)=\underset{t\to \infty }{lim}{F}_X\left(t\right)=1$$