Assignment 2 Solutions
March 24, 2014
1
1. x E n the norm of
x
x
is given by
1 2
x
x
x
x
x=
2
2
= 1.
Consider two vectors x, y E n . Now we will nd norm of sum of normalized x and y:
y
x
+
x
y
x
y
+
x
y
xx
yy
2x y
+
+
2
x
x y
y 2
xy
= 2 1+
x y
=
(1)
(2)
L
Assignment I Solutions
March 9, 2014
1. We have P (A|B) = P (A B)/P (B). But since A B we know that A B = A. Thus P (A B) = P (A)
and nally we get P (A) = P (A)/P (B)
2. Consider the expectation E (Xh(Y )|Y )
This can be wrtitten as E (Xh(Y )|Y ) =
xh(y)f
Assingment 3 Solutions
March 24, 2014
1
1. (a) (Dichotomized Regression) First consider E[i ]. In order to compute this expectation
x
we need to know the density of xi f (i ). But since xi is binary random variable it
x
takes only two values:
P (i = 1) =
Assignment 2 Solutions
April 5, 2014
1
1. Assume p(z) is a random variable. Then Fp (x) = P (p(z) < x). Bu this can be written
as:
P (2(1 (|z|) < x) = P (|z|) > 1
x
2
(1)
= P 1 (|z|) > 1 1
= P |z| > 1 1
x
2
= 1 P |z| 1 1
x
2
(3)
x
2
x
x
z 1 1
2
2
x