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1
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Chapter 3
Continuous Random Variables(R.V)
Part1
3
1 2 2 1
( ) [ ]
( ) 0 , ( ) 1
[ ] ( ) ( )
X
X X
X X
F x P X x
F F
P x X x F x F x
4
The CDF of X
Find:
0 1
1( ) 1 1
2
1 1
X
x <
xF x x <
x
[ 0.5] 1 [ 0.5]
1 (0.5)
0.5 1 1 0.25
2
X
P x P x
F
( ) [ 0.5]a P x
5
[ 0.5 0.75] (0.75) ( 0.5)
0.75 1 0.5 1 0.625
2 2
X XP x F F
(b) [ 0.5 0.75]P x
[| | 0.5] [ 0.5 0.5] (0.5) ( 0.5)
0.5 1 0.5 1 0.5
2 2
X XP x P x F F
(c) [| | 0.5]P x
6
1[ ] ( ) 0.8 0.6
2X
aP x a F a a
(d) [ ] 0.8 , Find a ?P x a
(e) the PDF ( )?Xf x
1 1 1( )
( ) 2
0
XX
x <dF xf x
dxotherwise
7
8
(1) 1XF
(1) [ 1] of shaded region=1XF P X area
1
(1) [ 1] ( )X xfF P x dxX
9
[ 0.5] area under the PDF=0.25P x
, Now Repeat part (a) P[x > 0.5]
10
area under the PDF=(0.75+0.5)*0.5=0.625
[ 0.5 0.75]P x Repeat part (b)
11
The CDF of V
Find:
2
0 5
( ) ( 5) 5 7
1 7
V
v <
F v c v v <
v
2 1(7 ) (7 ) 1 (7 5) =
144V VF F c c
Find the value of c ?(a)
12
2
[ 4] 1 [ 4] 1 (4)
1 1 (4 5) 0.4375
144
XP v P v F
(b) [ 4]P v
2
2 2 1 [ ] 1 [ ] [ ]
3 3 3
1 1 1[ ] ( 5)
3 144 3
1.92
V
P v a P v a P v a
F a a
a
(c) [ ] 2 / 3 , Find a ?P v a
13
2
0 5
1( ) ( 5) 5 7
144
1 7
V
v <
F v v v <
v
14
1( 5) 5 7
( ) 72
0 otherwis
( )
e
VV
dF v <v
v
dv
vf
15
[ 7] (7) 1VP v F
7 7
5
12( ) ( ) (0.5*12* ) 1 ]
7[ 7
2x xf x dx f x dP v x
16
The CDF of W
Find:
0 5
5 5 3
8
1( ) 3 3
4
3 31+ 3 5
4 8
1 5
W
w <
ww <
F w w <
ww <
w
3 4 31 5(4) +
4 8 8WF
P[W 4] ?(a)
17
1 1P[ 2 W 2] (2) ( 2) 0
4 4W WF F
P[ 2 W 2] ? (b)
18
1 1 [ ] [ ]
2 2
3 35 1we can substitute in : (1) or (2) +
8 4 8
5 5 1: (1) [ ] 1
8 8 2
, but 5 3 not valid
3 3 3 31 1 1: (2) + [ ] + 3.67
4 8 4 8 2
W
W
W
P W a F a
ww
w aFor F a a
w <
w aFor F a a
, where 3 5 valid
3.67
w <
a
[ ] 1/ 2 , Find a ?P W a (c)
19
2
1
1 2 2 1
( )( )
( ) ( )
( ) 1
[ ] ( ) ( ) ( )
( ) 0 for all x
Xx
x
X x
x
x
x X X
x
x
dF xf x
dx
F x f x dx
f x dx
P x X x f x dx F x F x
f x
: ( ) is a nondecreasing function its derivative ( ) is nonnegativeX xprove F x f x
20
The PDF of X
Find:
0 2( )
0 X
cx xf x
otherwise
2 22
0
0
( ) 1
| 2 12
1/ 2
xf x dx
xcx dx cxdx c c
c
Find the value of c ?(a)
21
1 1
0 0
1P[0 x 1] ( )
2 4x
xf x dx dx
P[0 x 1] ? (b)
1/2 1/2
1/2 0
1 1 1P[ x ] ( )
2 2 2 16x
xf x dx dx
1 1 P[ x ] ?
2 2 (c)
22
2
0
2
( ) ( ) 2 4
0 <
( ) 0 24
1 2
x x
X x
X
x xF x f x dx dx
x 0
xF x x
x
XFind CDF F (x) ?(d)
23
The CDF of X
Find:
0 < 1
1( ) 1 1
2
1 1
X
x
xF x x
x
0 < 11
1 1( ) 1( ) 1 1 2
20 otherwise
0 1
Xx
x
xdF xf x x
dx
x
Find the PDF ?
24
The PDF of X2 +bx 0 1
( )0 otherwise
X
ax xf x
1 1 1
2 2
0 0 0
( ) 1
+bx 1 1
31 3
3 2 2
xf x dx
ax dx ax dx bxdx
a ba b
Find the range of a and b ?
25
2 2
2 2 2
2
[ ] ( )
[ ] ( )
[ ] 0
[ ] [ ]
[ ] [ ]
[ ] [ ]
x
x
x
x x
E x xf x dx
E x x f x dx
E x
E aX b aE X b
Var X E X E X
Var aX b a Var X
26
The PDF of X
21/ 4 1 3
( ) , Y=h(x)=X0 otherwise
X
xf x
3 3
1 1
3 3
2 2 2 2
1 1
22 2
1[ ] ( ) ( ) 1
4
1 7[ ] ( ) ( )
4 3
7 4[ ] [ ] 1
3 3
x x
x x
x
E X xf x dx xf x dx x dx
E X x f x dx x f x dx x dx
Var X E X
Find the E[X] and Var[X] ?(a)
27
2
2
h(x)=X
h(E[X]) h(1)=1
7E[h(x)] E[X ]=
3
Find the h(E[X]) and E[h(x)] ?(b)
28
2
2
2 4 4
3 3
4 4
1 1
2
2 2
Y=X
7E[Y] E[X ]=
3
E[Y ]=E[X ]= ( )
1 ( ) 12.2
4
7[ ] [ ] 12.2 6.76
3
x
x
y
x f x dx
x f x dx x dx
Var Y E Y
Find the E[Y] and Var[Y] ?(c)
29
The CDF of X0 < 0
( ) 0 22
1 2
X
x
xF x x
x
2 2
0 0
0 < 01
0 2( ) 1( ) 0 2 2
20 otherwise
0 2
1: ( ) ( ) 1
2
Xx
x x
x
xdF xf x x
dx
x
check f x dx f x dx dx
Find the PDF ?(a)
30
2 2
0 0
2 2
2 2 2 2
0 0
22 2
3 3
1 0 2
( ) 2
0 otherwise
1[ ] ( ) ( ) 1
2
second moment:
1 4[ ] ( ) ( )
2 3
4 1[ ] [ ] 1
3 3
third moment:
[ ] ( )
x
x x
x x
x
x
xf x
E X xf x dx xf x dx x dx
E X x f x dx x f x dx x dx
Var X E X
E X x f x d
2 2
3 3
0 0
1( ) 2
2xx x f x dx x dx
Find [ ] , Var[X] , third moment?E X
31
The CDF of W
Find:
0 5
5 5 3
8
1( ) 3 3
4
3 31+ 3 5
4 8
1 5
U
u <
uu <
F u u <
uu <
u
E[ ] ?U(a)
32
3 5
5 3
3 5
5 3
1 5 3
8
( ) 3( ) 3 5
8
0 otherwise
1 3: ( ) 1
8 8
1 3[ ] ( ) 2
8 8
UU
U
U
u <
dF uf u u <
du
check f u du du du
E U uf u du udu udu
33
U E[2 ] ?(b)
3 5
5 3
1 5 3
8
3( ) 3 5
8
0 otherwise
1 3[2 ] 2 ( ) 2 2 13
8 8
U
U U U U
U
u <
f u u <
E f u du du du
