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Suppose our sample space is $\Omega = \{H,T\}$ and we have two random variables $X_1 , X_2$ such that $X_1(H) = 1,X_1(T) = 2$ and $X_2(H) = 3,X_2(T) = 4$. If we define stochastic process $S = \{X_1,X_2\}$ is $\{1, 4\}$ a realization of $S$ or only sets $\{1, 3\},\{2, 4\}$ are possible realization of process $S$? I though a realization means a sequence of sampling from each random variable in a process but reading this sample function definition (also page 9) seems i was totally wrong and $\{1, 4\}$ cant be realization of $S$ because realization is defined for a fixed $\omega \in \Omega$ and in the case of $\{1, 4\}$ the first outcome is $H$ and the second outcome is $T$.

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If $\omega=H$ then $X_1=1$ and $X_2=3$, and similarly if $\omega=T$ then $X_1=2$ and $X_2=4$. So indeed the only realizations of $S$ are $\{1,3\}$ and $\{2,4\}$.

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