This is a really stupid question but I really want to know why the following is true. I'm trying to prove something else and I need to know why that last bit is true.
2026-04-06 01:07:23.1775437643
If $E(X)=0$ then $P(X=0)=1$?
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No, Just look at the following p.m.f. $$ p(x)=\begin{cases} \frac{1}{2}, & \text{if $x=-1$} \\ \frac{1}{2}, & \text{if $x=1$} \end{cases}$$ Here, the expectation of $p$ is $0$ although $p(0)=0$.