Simple homework question about normally distributed variables The question states: 

Consider a set of random variables $X_i$, where $i=1,...n$. Each $X_i$ is
  normally distributed with mean $0$ and variance $1$, i.e. $X_i$ are $\mathcal N(0,1)$.
  What is the mean and the variance of the random variable $Y$, where
  $Y=X_1+...+X_n$.

How do I do this?
 A: Regarded as a question in probability theory, the answer to this question is that $$E[Y]=E[X_1+X_2+\cdots +X_n] = E[X_1] + E[X_2]+\cdots + E[X_n]$$ 
via a result known as the linearity of expectation, and since the random variables
all have zero mean in this particular instance, $E[Y]=0$.  On the other hand, 

the 
  information given is insufficient to determine the variance of $Y$. 

The variance of $Y$ is 
$$\operatorname{var}(Y) = \sum_{i=1}^n \operatorname{var}(X_i)
+ 2 \sum_{i=1}^{n-1}\sum_{j=i+1}^n\operatorname{cov}(X_i,X_j)$$
and so unless one knows (or makes assumptions about) the covariances, the variance
of $Y$ cannot be determined. One common assumption is that the $X_i$ are
independent random variables in which case the variance of $Y$ is just the
sum of the variances. The weaker condition that the $X_i$ are uncorrelated
random variables also leads to the same result, which in this instance is 
that $\operatorname{var}(Y)=n$.
The answers are different if one is talking about the sample mean and
the sample variance of $n$ samples from a standard normal distribution.
A: Working off Peter Flom's suggestion, look at this R code:
x <- as.data.frame(matrix(rnorm(50000,0,1), nr = 5000, nc = 10))
x$y <- x$V1 + x$V2 + x$V3 + x$V4 + x$V5 + x$V6 + x$V7 + x$V8 + x$V9 + x$V10

We generate 10 variables (V1 through V10), each with a mean of 0 and a standard deviation of 1.  Y is constructed to be the sum of the ten variables for each observation.  You can then run:
colMeans(x)
diag(var(x))

And those results should give you some insight about what to expect.
Then, look at this Wikipedia entry (the formula for Z specifically) and see if your insights match theirs! 
