probability - Cumulative distribution function determine the random variable

I don't know that determine is the right word, but I try to explain. What I need to understand. :) So..
We know's that if a function fit this conditions:

• Monotonically non-decreasing for each of its variables


• Right-continuous for each of its variables.

$$
0 \le F(x_1,\ldots,x_n) \le 1
$$
$$
\lim_{x_1,\ldots,x_n\to\infty} F(x_1,\ldots,x_n)=1
$$

$$
\lim_{x_i\to-\infty} F(x_1,\ldots,x_n) = 0,\text{ for all } i
$$
then the function is or can be a cumulative distribution function.

In this logic the cumulative distribution function determine the random variable? How I can prove it in mathematical way? This is true, I understand in my own way, but not mathematically.

Maybe we can start that the cumulative distribution function determine the probability distribution and vica versa. But how I can prove it mathematically that, the probability distribution determine random variable?

Thanks for your explanation,

I am really grateful:)

Answer

In general the CDF does not determine the distribution function. Consider for instance the uniform distributions over $[a,b]$ and over $(a,b)$. The distribution functions are different but it is straightforward to check that the CDFs are identical.