a. , , and each have mean 0, and by linearity of expectation we have
b. By definition of correlation, we have
where denotes the covariance,
Because are mutually independent, the expectation of their products distributes over the factors:
and recall that variance is given by
so that in this case, the second moment is exactly the variance of ,
We also have
and similarly,
So, the correlation is
c. The variance of is