Continuous Random Variables

Continuous Random Variable

An r.v. has a continuous distribution if its CDF is differentiable. We also allow there to be endpoints (or finitely many points) where the CDF is continuous but not differentiable, as long as the CDF is differentiable everywhere else. A continuous random variable is a random with a continuous distribution

Therefore, the PMF of continuous r.v.s is $0$ everywhere since $P(X=x)$ is the height of a jump in the CDF at $x$, but the CDF of $X$ has no jumps.

Probability Density Function

For a continuous r.v. $X$ with CDF $F$, the probability density function (PDF) of $X$ is the derivative $f$ of the CDF, given by $f(x)=F^{\prime }(x)$. The support of $X$, and of its distribution, is the set of all $x$ where $f(x)>0$

On the other hand, we also have

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

The PDF $f$ of a continuous r.v. must satisfy the following two criteria

• Non-negative: $f(x)\ge 0$
• Integrates to $1$: ${\int }_{-\infty }^{\infty }f(x)\mathrm{d}x=1$