# Continuous Random Variables

Suppose $X$ is some uncertain continuous quantity.

Then the probability that $X$ lies in any interveal $a \leq X \leq b$ can be computed as follows:

Define events $A = (X \leq a)$, $B = (X \leq b)$ and $W = (a \lt X \leq b)$

$B = A \cup W$

and $A$ and $W$ are mutually exclusive

This means that

$p(B) = p(A) + p(W)$

and so

$p(W) = p(B) - p(A)$

Now define the function $F(q) = p(X\leq q)$. This is called the **cumulative distribution function** or cdf.

Now, the **probability density function** is

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