Continuous Random Variables

Suppose XX is some uncertain continuous quantity.

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

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

B=AWB = A \cup W

and AA and WW are mutually exclusive

This means that

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

and so

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

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

Now, the probability density function is

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