Distribution Function

Topics: Probability

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(definition)

Let $X$ be a random variable. The distribution function of $X$ is a function $F_X:\mathbb{R}\to [0,1]$ such that:

$$F_X(x)=\mathbb{P}(X\le x)$$

Distribution functions have several properties. We can also define a distribution function at $x$ from the left.

Developed Form §

(theorem)

Notice that:

$$F_X(y)=\mathbb{P}(X\le y)=\sum _{x\in R_X}^y{f}_X(x)$$

Discrete §

With this, we can write, for $X$ a discrete random variable:

$$F_X(x)=\mathbb{P}(X\le x)=\sum _{x_i\le x}\mathbb{P}(X=x_i)$$

Continuous §

For $X$ a continuous random variable:

$$F_X(x)=\mathbb{P}(X\le x)={\int }_{-\infty }^x{f}_X(t)dt$$

Well-Defined §

(observation)

A distribution function is well-defined, since:

1. $(-\infty ,x]\in \mathcal{B}(\mathbb{R})$

2. $\{X\in (-\infty ,x]\}\in \mathcal{F}$

3. $\mathbb{P}(X\in (-\infty ,x])=\mathbb{P}(X\le x)=F_X(x)$

…with $\mathcal{B}(\mathbb{R})$ a Borel σ-algebra and $\mathcal{F}$ a σ-algebra.