Cumulative Distribution Function 10/5

Definition:
Let X be a continuous real-valued random variable, its cumulative distribution function is:

$$F_X(x)=P(X\leq x)$$

Theorem:
If X has a density f(x) then:

$$1. \qquad F(x)= \int_{-\infty}^x f(t) dt$$

is the cdf and

$$2. \qquad \frac{d}{dx} F(x)=f(x)$$

Proof:
Property 1. comes from the definition of probability as a function of the density:

$$P(\left (-\infty, x \right ])= \int_{-\infty}^x f(t)$$

.
Property 2 is due to the fundamental theorem of calculus.