# What is the mistake in this computation involving the absolute value of the expected value?

It is well known that $$|E[X]|\leq E[|X|]$$, where $$X$$ is a random variable taking values in $$\mathbb{R}$$ and $$E[X]$$ is its expected value.

Now, suppose that $$|E[X]| for some $$a\in \mathbb{R}$$.

This is equivalent to $$-a and multiplying by -1 we get $$a>-E[X]=E[-X]>-a$$.

If $$X\geq 0$$ then $$|X|=X$$ so that $$E[|X|]=E[X]. If $$X < 0$$ then $$|X|=-X$$ so that $$E[|X|]=E[-X].

In conclusion, this would show that $$|E[X]| implies $$E[|X|]. But since this works for any $$a\in \mathbb{R}$$ then $$|E[X]|< E[|X|]$$ would be impossible, violating the well-known inequality.

There must be some beginner's mistake in my elaboration and I can't seem to find it.

You are assuming that $$X \geq 0$$ with probability $$1$$ or $$X \leq 0$$ with probability $$1$$. Your argument fails if $$X$$ takes both positive and negative values with non-zero probability.