Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Jan 20, 2010 #1
    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.
    (a) What distribution does ||x||^2 follow? Justify your answer.
    (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. jcsd
  3. Jan 20, 2010 #2

    statdad

    User Avatar
    Homework Helper

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

    Once you know the distribution of [tex] |Z|^2 [/tex] you can answer the second question. Work on those before thinking about the third.
     
  4. Jan 20, 2010 #3
    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)???
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook