What function encodes the information we know about the result of an experiment?
What does it mean when we state that two random variables are identical?
Imagine an experiment performed in the future that generates one outcome from a set of mutually exclusive possibilities. We might describe the outcome from this experiment $X$using a number from the set of all real numbers $\mathbb{R}$. We cannot possibly know the value of this number before we do the experiment, however. Even so, we may know something so we define a function $F_X(x)$ called the cumulative probability distribution function, which encodes all the information we know about the likelihood of getting a particular result.
We do experiments and run simulations in order to extract information on the cumulative probability distribution function that we are sampling from when we do experiments. When others attempt to reproduce our results the hypothesis tests they perform are endeavouring to show that the distribution of results they obtain has the same distribution that we generated from our experiments.
Two random variables are thus said to be identical if they have the same cumulative probability distribution function.
<aside> 📌 SUMMARY: Random variables are used to describe the outcome of experiments. Two random variables are said to be identical if the distributions that they were sampled from are the same.
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