When constructing a random sample in Probability and Statistics, why doesn't the random sample all take on the same value?
For example, if $(X_1,\dots,X_n)$ is the random sample, why isn't the value $(X_1(\omega),\dots, X_1(\omega))$? Surely I must be missing something here. I thought the distribution characterizes the random variable so it should be that $X_1(\omega)=X_2(\omega)=\dots X_n(\omega)$. If $\omega$ is a person in a population, and $X_1(\omega)$ maps to the height of this person, wouldn't all o $(X_1(\omega),\dots, X_n(\omega))$ map to the height of that one person?