MATH 3081 Homework 10 .
Name:
3.7.1 pX,Y (x, y) = cxy at the points (1, 1), (2, 1), (2, 2), and (3, 1), and equals 0 elsewhere, find c.
((i 1) .
- ((2 1) ·
= C(2 2) . - (13 .
1) = I
9) = 1
O c=t
9
3.7.8 Consider the experiment of tossing a fair coin three times. Let X denote the number of heads on the last flip,
and let Y denote the total number of heads on the three flips. Find pX,Y (x, y).
Px g(x y&
, , O I
"
O C
I
is
·
2 Ys
3
⇢
c, 0 x 1, 0 < y < 1
3.7.10 Suppose that X and Y have a bivariate uniform density over the unit square: fX,Y (x, y) =
0, elsewhere
!'
(a) Find c
did =
Jodi .
Q
(b) Find P (0 < X < 0.5, 0 < Y < 0.25).
5
p(ocxx
!
.
5)
x)
:
1dx
.
= =.
5
·. 5dy :.
5yj .
, X
3.7.11 Let X and Y have the joint pdf
(x+y)
fX,Y (x, y) = 2e , 0 < x < y, 0 < y
-C
Find P (Y < 3X).
55
j(z dyd-n Jes (ediin
e
↳
Se(et3 erz ! 2 -Her
-e
-
dy :
2( 3 -
ep z(o =
-
( -
3.7.12 A point is chosen at random from the interior of a circle whose equation is x2 + y 2 4. Let the random variables
X and Y denote the x and y coordinates of the sampled point. Find fX,Y (x, y).
#
0
,
2
F2
1+
y = 22
A = TV2 =
4π
y(t y) ti
2 2
-
2 ,
o
fx
,
:
, ,
.. 2
0
3.7.17 Find the marginal pdfs of X and Y for the joint pdf derived in Question 3.7.8.
x 0123 o ↑
y
-3)3()
Px p(y) +
Name:
3.7.1 pX,Y (x, y) = cxy at the points (1, 1), (2, 1), (2, 2), and (3, 1), and equals 0 elsewhere, find c.
((i 1) .
- ((2 1) ·
= C(2 2) . - (13 .
1) = I
9) = 1
O c=t
9
3.7.8 Consider the experiment of tossing a fair coin three times. Let X denote the number of heads on the last flip,
and let Y denote the total number of heads on the three flips. Find pX,Y (x, y).
Px g(x y&
, , O I
"
O C
I
is
·
2 Ys
3
⇢
c, 0 x 1, 0 < y < 1
3.7.10 Suppose that X and Y have a bivariate uniform density over the unit square: fX,Y (x, y) =
0, elsewhere
!'
(a) Find c
did =
Jodi .
Q
(b) Find P (0 < X < 0.5, 0 < Y < 0.25).
5
p(ocxx
!
.
5)
x)
:
1dx
.
= =.
5
·. 5dy :.
5yj .
, X
3.7.11 Let X and Y have the joint pdf
(x+y)
fX,Y (x, y) = 2e , 0 < x < y, 0 < y
-C
Find P (Y < 3X).
55
j(z dyd-n Jes (ediin
e
↳
Se(et3 erz ! 2 -Her
-e
-
dy :
2( 3 -
ep z(o =
-
( -
3.7.12 A point is chosen at random from the interior of a circle whose equation is x2 + y 2 4. Let the random variables
X and Y denote the x and y coordinates of the sampled point. Find fX,Y (x, y).
#
0
,
2
F2
1+
y = 22
A = TV2 =
4π
y(t y) ti
2 2
-
2 ,
o
fx
,
:
, ,
.. 2
0
3.7.17 Find the marginal pdfs of X and Y for the joint pdf derived in Question 3.7.8.
x 0123 o ↑
y
-3)3()
Px p(y) +
