Are two standard normal random variables always independent?

I learned that the standard normal distribution is unique because the mean and variance are fixed at 0 and 1 respectively. By this fact, I wonder if any two standard random variables must be independent.

Answer

The answer is no. For example, if X is a standard random variable, then Y=X follows the same statistics, but X and Y are clearly dependent.

Attribution
Source : Link , Question Author : C.Hawk , Answer Author : zap

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