Question: I don't understand the following solution. Can I get a deeper explanation of the parts I highlighted? The first one is that I don't understand

I don't understand the following solution. Can I get a deeper explanation of the parts I highlighted?

The first one is that I don't understand why the first part is X instead of Y

The second one is that I don't understand how do we get X/2 here

The third one is that I don't understand why can that step be transferred to 1/2*E(X^2)

The forth one is that I don't understand why do we compute the first equation. Do we just compute a general uniform expectation here in which means E(x) = E(z), E(H) = E(z)

I don't understand the following solution. Can I get a deeper explanation

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