Expected Value: How to determine using random variables that these E(X) equations are equivalent or not?

Question

To determine for each scenario that if for any random variables X, Y, Z that these are true or false: All of them are false but how do I prove it? 3) looks like it should be true if I consider a coin toss scenario, same with 2). How are all these false??

1) $E(min(X,Y,Z)) = min(E(X), E(Y), E(Z)) $

2) $E(X.Y) = E(X).E(Y)$

3) $E(1/X) = 1/E(X)$

Answer 1

To see that they are false, all you need to do is provide a counterexample for each, ideally one for which the LHS and RHS can be easily computed.

Here are some suggestions:

1. Take $X, Y$ degenerate and both equal to zero with probability one. Can you think of a $Z$ which has expected value zero (so the RHS is zero) but for which the LHS is negative? (You can choose a coin toss.) Note this simplifies your general statement to $E(\min(0, Z)) = \min(E(Z)) = E(Z)$, which I hope you can see is clearly not true.

2. Equality holds when $X, Y$ are independent. Pick $X$ to be a coin toss with mean zero. This guarantees the RHS is zero. Can you think of a $Y$ for which the LHS is nonzero? (You can choose $Y$ to be a function of $X$.)

3. If $X$ is discrete and nonnegative and takes value zero with positive probability, what values does $1/X$ take?