We observe the discrete random variable $X = (X_1, . . . , X_n)$ with state space $S$, whose distribution we do not know but we are assuming that its joint p.m.f. belongs to a known family {${f_θ : θ ∈ Θ}$}. We derive a sufficient statistic $T(X)$.
Explain why it is wrong to write $E[X]$ and correct it.
Hey, I am unsure on this question, as I see nothing wrong with that.
If I had to guess at what "Explain why" is hinting at, it's that the expectation of $X$ depends on $\theta$. You know that $X$'s distribution is one member of the family $\{f_\theta\}$, but you're not told which member. Therefore to be specific you have to write $E_\theta[X]$, i.e., the expectation of $X$ assuming $\theta$ is true, and then continue your analysis with that assumption.