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 $W = (a \lt X \leq b)$

and $A$ and $W$ are mutually exclusive

This means that

and so

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