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Let say we flip a fair coin, the set of outcome is S={H,T}, P(H)=1/2, P(T)=1/2. Define random variable X:S->R by X(H)=1, X(T)=-1.

From what I read in books, I can define X1 and X2 as independent identically distributed (iid) random variables with the same distribution as X. Then, that would mean P(X1=1,X2=1)=P(X1=1)P(X2=1).

It can easily be seen that the event {X1=1,X2=1}={w in S:X1(w)=1,X2(w)=1}={w in S:w=H}={w in S:X1(w)=1}={X1=1}={w in S:X2(w)=1}={X2=1}, and since P({w:w=H})=P(H)=1/2, we have P(X1=1,X2=1)=1/2 and P(X1=1)P(X2=1)=1/4.

So, I am not sure exactly what I mistake I made. Please help me clear up my confusion. Thanks.