Find density function $f_{X}$ of $X$

Question

Distribution function of random variable $X$ is:
$$F(x) = \begin{cases} 0 \mbox{ for } x<0 \\ x \mbox{ for } x \in (0,1) \\ 1 \mbox{ for } x \ge 1 \end{cases} $$ Find density function $f_{X}$ of $X$.

I am taking first steps with continuous random variables. It seems that $F(x)$ represents some kind of uniform distribution (continuous). I know that $P(x \in A) = \int_A f_X(x) dx $ and $F_X(x) = P(X \le x)$ but how to use that in given simple case?

Answer 1

$f_X$ is simply the derivative of $F$. So $f_X(x)=1$ for $0<x<1$ and $0$ elsewhere. [This is the uniform distribution on $(0,1)$].

Answer 2

The relation between the distributon function $F_X(x)$ and the densitiy function $f_X(x)$ is given by $F_X'(x)=f_X(x)$ in points of where $F_X$ is differentiable.

That means $$f_X(x)=1_{(0,1)}$$ which is a uniform distribution.