# Homework Help: Applied Stats Help - Don't even understand the question

1. Jan 20, 2010

### jimbodonut

1. The problem statement, all variables and given/known data

Suppose x = (x1, x2, ..., xn)T ∈ Rn is a random vector drawn from the n-dimensional
standard Gaussian distribution N(0, I), where 0 = (0, 0, ..., 0)^T (0 vector transpose) and I is the identity matrix.
(b) On the average, how far away (in terms of squared Euclidean distance) from the
origin do you expect x to be? In other words, what is E(||x||^2)?
(c) Now suppose we fix x and draw another random vector z from N(0, I). If we
project z onto the direction of x, how far away from the origin (again, in terms of squared
Euclidean distance) do you expect the projection to be? (Hint: Let u be the unit vector
pointing in the direction of x. Then, uT z is the projection of z onto the direction of x.
Find the expectation and variance of uT z conditional on u.)

2. Relevant equations

3. The attempt at a solution

no attempt... don't even understand the question... :P

Thanks guys... ur help is greatly appreciated...

2. Jan 20, 2010

Start with the univariate case: if Z is normal 0, 1, what do you know about the distributoin of Z^2?
If $$Z \sim \text{MVN}_n \big(0, I\big)$$, how does the univariate case relate to $$|Z|^2$$ in the multivariate case?

Once you know the distribution of $$|Z|^2$$ you can answer the second question. Work on those before thinking about the third.

3. Jan 20, 2010

### jimbodonut

i figured out a) and b).

It is a chi-square distribution since the ||x|| = sqrt(x1^2 + x2^2 + ... + xn^2) thus ||x||^2 = x1^2 + x2^2 + ... + xn^2. Since xi~N(0,1), it is chi-square.

and the expectation of a chi-square distribution is its degrees of freedom... in this case... E(||x||^2) = n

but now what's c)???