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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:)

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    I understand now, thank you, please write an answer and I accept it, because you are the first.2012-11-05

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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.

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    thank you I understand now, this is a good example to. )2012-11-05