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- The distribution function
The distribution function
The distribution function of a random variable $X$ is a function that assigns, for every point, the probability accumulated up to the above mentioned value. That is:
$$F(X)=p(X \leq x)$$
For example, we will compute the distribution function of the random variable $X$, resulting from throwing a perfect dice.
The following table shows the values of $F (x)$:
| x | F (x) |
| $x < 1$ | $0$ |
| $1\leq x < 2$ | $\displaystyle \frac{1}{6}$ |
| $1\leq x < 3$ | $\displaystyle \frac{2}{6}$ |
| $1\leq x < 4$ | $\displaystyle \frac{3}{6}$ |
| $1\leq x < 5$ | $\displaystyle \frac{4}{6}$ |
| $1\leq x < 6$ | $\displaystyle \frac{5}{6}$ |
| $x \leq 6$ | $1$ |
The value of the function distribution in $-\infty$ will always be $0$, while the value in $+\infty$ will always be $1$.
This turns out to be quite intuitive, since the probability that the value of $x$ is smaller than $-\infty$ is zero, and the probability that it is less than $+\infty$ is $1$ (since it is always less or equal to $+\infty$).
Since it is a discrete random variable, the distribution function will be discrete. Note that the probability of obtaining a result lower than $5,2$ is the same as that of lower than $5,3$ or $5,9$.