What is lower bound for expected value of sum of iid random variables?

Question

Let $x_1,…,x_n$ be iid positive random variables where $x_i \in \{1,2,...,k\} $ and $E(x_i)=\mu$, and $\mu$ is unknown, now assume $X=x_1+x_2+...+x_n$, we know from central limit theorem that $E(X)=n \times \mu$, but as i said $\mu$ is unknown, can we have an upper bound or lower bound for $E(X)$?

Edit

all of $k$ values for $x_i$ has a chance to be chosen. For example we dont have $Pr(x_i=5)=0$ where $5<k$

Answer 1

can we have an upper bound or lower bound for $E(X)$?

Yes, we can!

Clearly $n\leq \mathsf E(X)\leq kn$

Edit all of $k$ values for $x_i$ has a chance to be chosen.

All that tells us is that the extremes are not plausible values of $\mathsf E(X)$, so$$n<\mathsf E(X)<nk$$