Problem 1.

Let

X, Y, Z

denote 3 jointly distributed continuous random variables with joint density

function given by

f

(

x, y, z

) =

{

c

(

x

2

+

yz

) 0

x

1

,

0

y

1

,

0

z

1

,

0

otherwise

Using the above joint pdf, answer the following:

1a. Find

c

.

1b. Find the marginal pdf of

X

1c. Find the bivariate marginal pdf of (

X, Y

).

1d. Find the conditional pdf

f

(

Z

|

X

=

x, Y

=

y

)

1e. Find the conditional pdf (

Y, Z

|

X

=

x

)

1f. Compute

E

(

XY Z

)