Distributions

Let $X$ be a random variable. We can determine a function $F_X :$   $\rightarrow$    (called the cumulative distribution - CDF - or simply distribution function of $X$) given by

$$F_X(x) := P(X \le x).$$ (1.4)

Clearly if $x < y$ then $(X \le x) \subset (X \le y)$ and hence $F_X(x) \le F_X(y)$, that is, $F_X$ is monotonically increasing. Furthermore, when $x \rightarrow +\infty$ the event $(X \le x)$ tends to $\Omega$ and when $x \rightarrow -\infty$ the event $(X \le x)$ tends to $\emptyset$. Hence it is reasonable that

$$\lim_{x \rightarrow -\infty} F_X(x) = 0, \quad \lim_{x \rightarrow +\infty} F_X(x) = 1.$$ (1.5)

The plots of cumulative distribution functions have the general form shown in figures 1.1 or 1.2.

Figure 1.1: A continuous CDF

Figure 1.2: A discrete CDF

How can we determine the cumulative distribution function of a given random variable $X$? We can sample values of $X$, repeating $N$ times the corresponding experiment, obtaining for each $x \in$    the table of results:

Repetition	Observation	$x_i \le x$?
1	$x_1$	yes
2	$x_2$	no
&vellip#vdots;	&vellip#vdots;	&vellip#vdots;
$N$	$x_N$	yes

Clearly

$$F_X(x) \simeq { {\char93 \{ i \le N : x_i \le x \} } \over N}$$

if $N$ is large, according to the frequency approach. The function $F_e$ given by

$$F_e(x) := { {\char93 \{ i \le N : x_i \le x \} } \over N},\;\;\; x \in$$   

is called the empirical cumulative distribution function of $X$. If $x_1 \le \ldots \le x_N$, it has a plot of the form represented in figure 1.2.