2.x ESL Chapter Two exercise 2.4

Topic Preparation

• $x _i\sim N (0,1) $, with $\sum_i^n x_i^2 \sim \chi^2 (N) $
  Where $n$ is called the degree of freedom, the mean value of chi-square distribution is its degree of freedom
• $x _i\sim N (\mu_i,\sigma_i^2) $, with $\sum_i a_ix_i \sim N (\sum_i a_i\mu_i,\sum_ia_i^2\sigma_i^2) $
  The linear and the N normal distribution variables still conform to the normal distribution
• Calculate the length of vector b projected onto vector x T, $t =|b|cos\theta=|b|\frac{x^tb}{|x| | b|} =\frac{x^t}{|x|} b=a^tb$
  So a is a unit vector $\sum_i a_i^2=1$

Exercises

$z =a^tx$, $Var (z_i) =\sum_j A_j^2var (x_j) =\sum_j a_j^2=1$<br>
So $z_i\sim N (0,1) $<br>

The distance squared $d^2\sim \chi^2 (p) $, so the average distance is $\sqrt p$

2.x ESL Chapter Two exercise 2.4