Let U and V be random variables. show that (E ( UV )) 2 E ( U 2 ) E ( V 2 ) (This is the Cauchy-Schwarz inequal- ity). Hint:

Let U and V be random variables. show that (E (

UV

))

2

≤

E (

U

2

) E (

V

2

)

(This is the Cauchy-Schwarz inequal-

ity).

Hint: Show that if

U

+

tV

= 0

for some constant

t

, then

(E (

UV

))

2

= E (

U

2

) E (

V

2

)

.

Otherwise, note that

0

<

E

(

(

U

+

tV

)

2

)

=

q

(

t

)

,

for all

t

,

where

q

is a quadratic polynomial in

t

. Since

q

has no zeros, its discriminant must be

negative.

(b) Suppose that

U

=

tV

+

b

, with

t

6

= 0

. Show that

ρ

U,V

= 1

if

t >

0

and

ρ

U,V

=

−

1

if

t <

0

.

(c) Show that if

X,Y

are random variables, then

|

Cov (

X,Y

)

|≤

√

Var (

X

)

√

Var (

Y

)

,

and conclude that

|

ρ

X,Y

|≤

1

. In fact,

|

ρ

X,Y

|

= 1

only when

Y

=

tX

+

b

with

t

6

= 0

.

Hint: Apply (a), with

U

=

X

−

E (

X

)

and

V

=

Y

−

E (

Y

)

.

2