Distribution Functions (cdf's)

The density or probability function is an idealised pattern which would be a reasonable approximation to represent the frequency of the data; the slight imperfections can be disregarded. If we can accept that approximation, we can reduce the data and understand it. To use the density or probability function, we usually have to integrate (or sum if it is discrete). The distribution function arises as the integral or sum. Whether we refer to the distribution or density (probability) function, we are still referring to the same information.

Read HC 1.7 where distribution functions for the univariate case are considered.

Example 2..2

Given random variables $X$ and $Y$ which are identically distributed and independent (iid), with pdf $f(x)$, $x>0$, find $P(Y>X)$. Consider one particular value of $Y$, say $y^\ast$. Then the probability that this value is greater than any $X$ is written mathematically as

$$P(y^\ast > X) = P(X < y^\ast) = \int_0^{y^\ast} f(x) dx \ \ .$$

Now to generalize for all $Y$, we need to take into account the frequency of $y^\ast$ and that information is contained in the density $f(y)$. We integrate the above probability over $f(y)$.

The joint pdf of $X$ and $Y$, $f_{X,Y}(x,y)$, can be written $f(x)f(y)$ so

$$\begin{eqnarray*} P(Y>X) & = & \int^\infty_0\!\int^y_0 f(x)f(y)\,dx\,dy \mbox{ ... ...c{\{F(y)\}^2}{2}\right]^\infty_0\\ & = & \textstyle{\frac 12} \end{eqnarray*}$$