Please include your name ( with your last name underlined ) and your NetID at the top of the first page.
No credit will be given without supporting work.
1 - 2. Let X and Y have the joint probability density function
f X , Y ( x, y ) =
x
1
, x > 1, 0 < y <
x
1
, zero elsewhere.
1. a) Let U = X Y. Find the p.d.f. of U, f U ( u ).
b) Let V = Y / X. Find the p.d.f. of V, f V ( v ).
2. Let U = Y and V = Y / X.
Find the joint probability density function of ( U, V ), f U, V ( u, v ).Sketch the support of ( U, V ).
3 - 5. Suppose that X and Y are independent,
X has a Uniform distribution on interval ( 0, 3 ),
and the p.d.f. of Y is
f Y ( y ) = y/2 , 0 < y < 2, zero otherwise.
3. Find the probability density function of W = X + Y, f W ( w ) = f X + Y ( w ).
4. Find the probability density function of V = X × Y, f V ( v ) = f X × Y ( v ).
5. Find the probability density function of U = Y/ X , f U ( u ) = f Y / X
( u ).
6 - 8. Consider two continuous random variables X and Y with joint p.d.f.
f X, Y ( x, y ) = x y
3
3
4
, 0 < x < 1, 0 < y < 3 x, zero otherwise.
6. a) Find P ( Y > 1 | X =
3
2
). b) Find E ( Y | X = x ).
c) Find P ( X <
3
2
| Y = 1 ).
7. d) Let W = X + Y. Find the p.d.f. of W, f W ( w ) = f X + Y ( w ).
8. e) Let U = X and V = X × Y.
Find the joint probability density function of ( U, V ), f U, V ( u, v ).
Sketch the support of ( U, V ).
f) Use part (e) to find the probability density function of V = X × Y, f V ( v ).
9 - 10. Let X and Y have the joint probability density function
f X , Y ( x , y ) = 10 x y
2, 0 < x < y < 1, zero otherwise.
9. Let U = Y - X and V = X + Y.
Find the joint probability density function of ( U, V ), f U, V ( u, v ).
Sketch the support of ( U, V ).
10. Let U = Y / X and V = X × Y.
Find the joint probability density function of ( U, V ), f U, V ( u, v ).
Sketch the support of ( U, V ).
11. Let X denote the number of times a photocopy machine will malfunction: 0, 1, 2, or 3 times, on any given month. Let Y denote the number of times a technician is called on an emergency call. The joint p.m.f. p ( x, y ) is presented in the table below:
x
p Y y 0 1 2 3 ( y )
0 0.15 0.30 0.05 0 0.50
1 0.05 0.15 0.05 0.05 0.30
2 0 0.05 0.10 0.05 0.20
p X ( x ) 0.20 0.50 0.20 0.10 1.00
a) Find the probability distribution of W = X + Y.
b) Find the probability distribution of V = X × Y.
12. 2.2.2 ( 7th and 6th edition )
Let X 1 and X 2 have the joint pmf p X 1 , X 2
( x 1 , x 2 ) = x 1 x 2/36 , x 1 = 1, 2, 3 and
x 2 = 1, 2, 3, zero elsewhere. Find first the joint pmf of Y 1 = X 1 X 2 and Y 2 = X 2 ,
and then find the marginal pmf of Y 1 .
Hint: X 1 and X 2 are discrete random variables. There are nine possible pairs ( x 1 , x 2 ).
If you are registered for 4 credit hours: ( please put the solution to these problems on the last page of your homework paper )
13. 2.3.6 ( 7th and 6th edition ) Let the joint pdf of X and Y be given by
( ) ( )
< < ¥ < < ¥
+ + =
0 elsewhere.
0 , 0
1
2
,
3
x y
f x y x y
(a) Compute the marginal pdf of X and the conditional pdf of Y, given X = x.
(b) For a fixed X = x, compute E ( 1 + x + Y | x ) and use the result to compute E ( Y | x ).
14. Let the joint pdf of X and Y be given by
( ) ( )
< < ¥ < < ¥
+ + =
0 elsewhere.
0 , 0
1
2
,
3
x y
f x y x y
a) Let W = X + Y. Find the p.d.f. of W, f W ( w ) = f X + Y ( w ).
b) Let V = Y / X. Find the p.d.f. of V, f V ( v ).