1. Two fair dice are rolled. Let $X$ be the outcome of one die and $y$ denote the sum of the dice. The joint and marginal probabilities of $X$ and $y$ are given by the table: (a) Using the marginal probability masses $p_{Y}\left(y_{j} ight) $, compute the mean $\mathbb{E] [Y]$. (b) For each $x_{i}=1,2,\ldots, 6$, compute the conditional expectation $\mathbb{E}\left( \mid X=x_{i} ight). (c) Using your answers to the previous part, define the random variable SELY \mid X]$. (d) Now using the definition of the expectation of a discrete random variable, compute $\mathbb{E} |\mathbb E) YIXI \mid .$ (Hint: If you have answered parts $a$ and $(d) $ correctly. your answers should be the same.) SP. SO. 023