Science:MATH105 Probability/Lesson 2 CRV/2.02 CDF Properties

CDFs share many of the same properties for both the continuous and discrete cases. In the following theorem, the three properties listed are common to both the discrete and continuous cases.

Theorem: Properties of a Distribution Function
If F(x) is a cumulative distribution function for the random variable X, then $\lim _{x\rightarrow -\infty }F(x)=0$ $\lim _{x\rightarrow +\infty }F(x)=1$ F(x) is a non-decreasing function of x

It may help the reader at this point to recall the definition of a "non-decreasing function": a function, f(x), is a non-decreasing function if f(x₁) ≤ f(x₂) for all x₁ and x₂ in the domain of f, and x₁ < x₂.

Proofs for the first two property are beyond the scope of MATH 105. For the interested reader, a proof of the third property is below.

Proof of The Third Property

In general, for any interval of finite length Δx, there is a non-negative probability that X lies somewhere in that interval, no matter where our interval lies or how small we make Δx. Therefore, if we were to choose any interval [x , x + Δx], the probability that our continuous random variable X lies inside of this interval,

${\text{Pr}}(x\leq X\leq x+\Delta x)$

must be non-negative:

$0\leq {\text{Pr}}(x\leq X\leq x+\Delta x)$

But this can be expressed as the difference

${\begin{aligned}0&\leq {\text{Pr}}(x\leq X\leq x+\Delta x)\\&={\text{Pr}}(X\leq x+\Delta x)-{\text{Pr}}(X\leq x)\\&=F(x+\Delta x)-F(x)\end{aligned}}$

Rearranging yields, for all ∆x,

$F(x)\leq F(x+\Delta x)$

This is the definition of a non-decreasing function. Therefore, the CDF, which is defined as F(x) = Pr(X ≤ x), is a non-decreasing function of x.