I have a problem as follows. As of now, I cannot provide the definition of X and Y but can anyone provide a rough overview of what needs to be done ?
An experiment consist of rolling a single die and the experimental outcome determines a value for X and Y. (the random variables X and Y are defined Table)
a) Find a constant a that will give a mininmum mean-square estimate of X and calculate the resulting mean-square error.
b) Given that Y=4, find the minimum mean-square estimate of X and the resulting mean-square error
The mean of a distribution is the estimate that minimizes the mean square error.
For example, imagine that we are going to draw $x$, and it can be $0$ or $1$ with equal (50%) probability. We make an estimate $\hat{x}$ to minimize the mean square error:
$$ \min_\hat{x} \ E\left[(x-\hat{x})^2\right] $$
where the $E$ means expected value.
Using calculus, the mean square error is minimized by a choice of $\hat{x}$ that satisfies:
$$ 0 = E\left[-2\left(x-\hat{x}\right)\right] $$
Dividing both sides by $-2$ and rearranging, we get
$$ \hat{x} = E\left[x\right] $$
So we should guess $\hat{x} = 0.5$ in this case.
For your problem, in part a), the answer will be the expected value of $X$, and in part b), the answer will be the expected value of $X$ conditional on $Y=4$.