I have understood that random variable let say X is a function that maps the all the outcomes of some particular random experiment with some real number according to some particular relation. But I am not able to understand the meaning of probability of random variable.(From my understanding probability of random variable should mean probability of some function, this is my guess actually, this is what I am able to comprehend from the texts that I have read so far. ) What does P(X = a) means , here "a" is some real number. Could somebody please explain me in detail the actual meaning of Probability of random variable in simple language and also the formal definition of it. If possible please provide the source of information you are using for your explanation. Thank you!
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$\mathbb{P}(X=a)$ is the probability of the set ${\omega\in\Omega: X(\omega)=a}$. If you pick a random element from the sample space, that is the probability that this element is mapped by $X$ to $a$. While a random variable is indeed a function, most of the time these probabilities (the distribution of $X$) are much more important than the actual values $X(\omega)$. – Mark Mar 20 '24 at 02:18