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(ELEC2600)[2013](s)midterm~htang^_12993.pdf
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ELECC210 Springg 2011 Midteerm Answerr Sheet
Sectioon bbcbd
Sectioon Q1. ..
a) P[3. X. 6] =P[X=4 or 5 or 6] ==
..= .
.
= .[... .. . ... .] .
P[1.X. 6|X.2]=PP[X=3or 44 or 5|X.2] .[...] = ......=...b)) The cdf of XX

.X)
.XX.)
3.X.
kp..
.....
.
.E.
)X..
k)
...X
=

c) E
E

..
. = 4.375
..
=

...
)=
=EX

)
.
.
)
= .... =
therefore t
.
[.
.
D
P

=

]

d))

7

hhe conditionaal3
4
5
i

isted as folllows,

.
.,
K 1 1 1 1 1
PP...|.) 3 6 112 4 6

e)) The probabbility of Y

Y 13 26 34 43
PP...) 16 16 16 16

.
E.Y) = .... , E.Y)
=

.9961

=

,

pmf of Xss l6
7
.....
D.Y) E.Y) .E.YY)
. = ...00
.. ...=
Q2.
a)) Total probaability is equual to 1, thereefore we havve the area u nder the funnction f.x) is 1.
.c+cc.2)1.2=1
4
Then c= 55.

....
b)) P[1.X. 3]=1. P[00.X.13.X.5]=1. 1 ..5.33) =
.. ...
..
x, 0.x. 2
..
.

c) The functioon f.x) =., 2 . x . 5, therefoore
.
00, otherwisse
0, ..0
.
, 0...
.

1
.

1
6
..1 +

2

.

1
1.
4..

F.X)
=

.), 2...55..
4

1,


d))

Expectatio
.
)X.
.
.
.
. ..
.+ ....
.
).
E

..

.

.
.

.
.. ..
..
..
....
.+ .... . =....
..
+

=

=

==

..
.
)
=E

).E.
E

X

.

.

.

..

..

.

..

..

.= .....
.
.
...

... ..
..
..
..
.+ .... = ...
. ..
..
..
.
.
.
+

+

=

==

=

.
)P[X. 1] .
.
..

)
1.

)X.
D

X

X

= ...
.. ........=1.
=

1.4983

.
.
P[

]

e))

11

...
= ....
2....1,.2.

154, 2....5
.........
0,
==

=

.x|A)
f

. .

.

.

==

Q3.
a)) 0.9 0.7 0.5 = 0.3155
b) 0.90.7=0.63 c) Let us define the event get the $100 as A and the event answer the question 2incorrectly as B, then
.
P[A]=1. [A]=1.0.315=0.685 P=0.9 .1.0.7)
.
[.].
A [.
= .

..
]]
.
, therefore P[A]
$P[$] = 2.7 + 6.3 + 31.5 = 40.5
P

[
=

B]
P

= 0.27
0.27
0.685 = 0.3942
.

[
P


]

[AA

.

P[B|A]Since B is a subcase of A.
=

[

]

.

=P

.

d) The probability of $
$....

[$] .$
=]$[E
$..

Q4.
a) ...... = ...
... ...b) ...
... ...
Vo
P
0 10 20 100
0.1 0.27 0.315 0.315

...
...+......=...
... ......for Ce

[

c) P
=

2]
for C2

te
Vo


[
= 0.2

|A group]P[A group] +P[Vote for C2|B group]P[B group]
+ 0.9 ..= ....
t

..

d) Accor
P

ing to Bayes Theorem in Group B|vote for C1]C1........ .0.8.
.....
d


.
........ ... ..
]
= ./...../.. = ..

[
[

= .[.. ..... . & .... ... ..]
.[.... ... ..]
0.2..... ....0.9.....
=

...[
]=


e) P

.

. 0.1
...

.
.