There are two random variables:
$X ={}$ the sum of $1,000,000$ real numbers
$Y ={}$ the same sum as $X$ with each number rounded to the nearest integer before summingIf the fractions rounded off are independent and each one is uniformly distributed over $\boldsymbol{(-0.5, 0.5)}$, use the CLT to estimate the probability that $|X-Y|>700$.
My thought process so far is that $X \thicksim \operatorname{Uni}(-0.5, 0.5)$ and $Y \thicksim \operatorname{Uni}(-1, 1)$, but since $Y$ depends on $X$, what would my actual definition of $Y$ look like? How do I figure out the difference between a uniform and discrete variable?
I think I understand what I am supposed to do once I figure out the correct normal variable to use the CLT on, but I'm just not quite sure how to get there. Any help is appreciated.
From my understanding of the question, I would set it up like this: we have $\{ X_i \}_{i=1}^{1000000}$ fixed deterministic numbers, which we sum $X = \sum_{i=1}^{1000000} X_i$, and then we wish to compare that with the sum $Y = \sum_{i=1}^{1000000} Y_i$ is, where $Y_i = \left[ X_i \right]$. We make the modeling assumption that the errors are a random variable $Z_i = Y_i - X_i \sim U(-0.5,0.5)$. Now we want to calculate $$P(|Y - X| > 700) = P(|\sum_{i=1}^{1000000} Y_i - X_i| > 700)$$ $$= P(|Z| > 700)$$ where $Z$ is a sum of one million uniform random variables and should obey the CLT. From here, we use our knowledge that $Z$ behaves like a normal distribution with mean 0 and variance $\frac{1000000}{3}$.
Hope that helps.