Probability Density Function

We say that X is a continuous random variable if there exists a nonnegative measurable function $f(x)$, defined for all real $x\in (-\infty ,\infty )$, having the property that for any (measurable) set $B$ of real numbers

$$P(X\in B)={\int }_{B}f(x)\mathrm{d}x.$$

The function $f(x)$ is called the probability density function of the random variable $X$.

Or equivalently, by the equivalence between CDF and law, a r.v. $X$ is continuous iff it possesses a density $f$ such that

$$F(t)={\int }_{-\infty }^{t}f(x)\mathrm{d}x.$$

Any nonnegative measurable function $f(x)$ such that

$${\int }_{-\infty }^{\infty }f(x)\mathrm{d}x=1$$

is called a density function. Obviously, a density function defines a Distribution Function that satisfies the Properties, and there exists a Random Variable whose PDF is the given density function.

Remarks

• The PDF of a continuous r.v. is not unique. But two PDFs are equal a.s.
• A continuous r.v. need not to be a continuous function.
  • A continuous r.v. has a continuous CDF.
  • However, any plateau in the CDF implies a jump (discontinuity) in $X$.
    • $X(\omega )=\omega +1\{1/2<\omega \le 1\}$ for $\omega \in [0,1]$. This is a non-continuous function with density $f(x)=1\{x\in [0,1/2]\cup (3/2,2]\}$.