Let $\bf{S}$ be a discrete random variable with set of outcomes $S=\mathbb{Z}_7$. Let $\bf{R}$ be a uniform discrete random variable with set of outcomes $R=\mathbb{Z}_7$ that is independent of $S$. For $i\in\mathbb{Z}_7^*$, let $\bf{U_i}$ be the discrete random variable representing the outcome of $i\bf{R}+\bf{S}$.
I am having trouble with the following:
Suppose $i\in\mathbb{Z}_7^*$. Find $H(\bf{U_i}$) and determine under what conditions $H(\bf{U_i}$)$=H(\bf{S})$.
So $S=\{0,1,2,3,4,5,6\}$ and $R=\{0,1,2,3,4,5,6\}$? So we want to find Pr($i\bf{R}+\bf{S}$) which I'm struggling to see what it is, I have produced a sum table over mod 7 but still can't see what it could be. Then I think we want to sub into the entropy formula $-\sum_{i=1}^nPr(i\bf{R}+\bf{S})\log$$(Pr(i\bf{R}+\bf{S})$? (with log base 2). Then I do not really know where to go from here or how to find the conditions st. $H(\bf{U_i}$)$=H(\bf{S})$. Any advice?