For random variables $X_1, ..., X_N (N = 100)$ such that each of their means are $0.1$ and $Y = \frac{1}{N}\sum_{i = 1}^NX_i$, how is the mean of $Y$ meant to be calculated? I don't really understand what $Y$ represents here; what does a sum of variables represent?
My intuition tells me that the expected value of $Y$ is simply the mean of means, which would thus be $0.1$, but I must be missing something conceptually because it's not all clear to me. Any help is appreciated, thanks!
The variable $Y$ is some random variable because it is a sum of some other random variables. To understand this conceptually: Imagine that you were rolling $N$ dice, with the outcome of each rolled die be $X_1,\cdots,X_N$ and the average of all the outcomes of the rolls were $Y$. Clearly you wouldn't know the average value $Y$ unless all the dies were revealed to you. So $Y$ is some random variable, at least conceptually. Get it?
As $Y$ is a random variable, it must have some probability distribution, but it is often unclear exactly what distribution it is, unless we knew specific things about the properties of the $X_i$. In this case you know that they all have expected value $0.1$, and as $Y$ is also a random variable it must have some expected value, which is most easily calculated using the "linearity property" of expected value, namely that $E(aX_i + bX_j) = aE(X_i) + bE(X_j)$ if $a,b \in \mathbb{R}$. By using this property multiple times you get $$E(Y) = E (\frac{1}{N}\sum_i^N X_i) = \frac{1}{N} \sum_i^N E (X_i)$$ Now since you know that $E(X_i) = 0.1$ we must have $E(Y) = \frac{1}{N}\sum_i^N 0.1 = \frac{1}{N}N 0.1 = 0.1$.