Cross Validated Asked by natalia on December 24, 2020

So, if $X$ and $Y$ are both continuous random variables with the same cdf, does that mean that their expectations are the same? And the same thing in case $X$ and $Y$ are both discrete.

Thanks in advance!

*Cumulative Distribution Function* (CDF) characterizes exhaustively any random variables. So if two rv's share the same CDF these are identical. Then yes, they are the same expected value if it exist. We can say more, these two rv's share all moments.

More interesting to study the situation in which two rv's share some moments. In this case is not sure that the them are identical (same CDF).

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