Let $X$ be a random variable with the p.d.f. $P\{X=x_i\}=p_i=\frac{1}{N}$ for $i=1,2,\dots,N$ and suppose that $N$ is an even number.
Let $X_1,X_2,\dots,X_m$ be i. i.d. variables distributed as $X$. Let $Y=\sum\limits_{k=1}^mX_k$.
What are the following:
i) $p_o=\sum\limits_{j=M, \\\text{j is odd}}^{MN}y_j$
ii) $m_o=\sum\limits_{j=M, \\\text{j is odd}}^{MN}jy_j$
Solution to i) Since $N$ is even, we consider two cases.
If $M$ is odd, $MN-M+1$ is even, so there are half as many odd numbers as there are even, and $p_o=\frac{1}{2}$.
If $M$ is even, $MN-M+1$ is odd, so we have only $\frac{MN-M}{2}$ odd numbers, and $p_0=\frac{MN-M}{2(MN-M+1)}=\frac{1}{2}\frac{1}{1+\frac{1}{M(N-1)}}$.
But I have no ideas how to caluclate ii)