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STAT213StatisticalInference-Assignment1
August17,2021
Question1
Foreachofthefollowing,determinewhetherthegivenfunctioncanserveastheprobabilitydistribution
ofarandomvariablewiththegivenrange:
a)
f
(
x
)=
x
2
5
for
x
=1
;
2
;
3
;
4
;
5
f
(1)=
1
2
5
=
1
5
<
0
Notaprobabilitydistribution.
b)
f
(
x
)=
x
2
30
for
x
=0
;
1
;
2
;
3
;
4
f
(
x
)=
x
2
30
0
;
8
x
=0
;
1
;
2
;
3
;
4
4
X
x
=0
x
2
30
=
0+1+4+9+16
30
=1
f
(
x
)
canserveasaprobabilitydistribution.
c)
f
(
x
)=
1
5
for
x
=0
;
1
;
2
;
3
;
4
;
5
f
(
x
)=
1
5
0
;
8
x
=0
;
1
;
2
;
3
;
4
;
5
5
X
x
=0
1
5
=
1+1+1+1+1+1
5
=
6
5
>
1
Notaprobabilitydistribution.
Question2
Theprobabilitydensityfunctionofarandomvariable
X
isgivenby
f
(
x
)=
Ëâ€
kxe
x
2
for
x>
0
0
for
x
0
Find
a)thevalueof
k
Substitute
u
=
x
2
and
du
=2
xdx
1=
Z
1
0
kxe
x
2
dx
=
k
Z
1
0
1
2
e
u
du
=
k
2
e
u
1
0
=
k
2
k
=2
b)thedistributionfunctionof
X
Z
x
0
2
xe
x
2
dx
=2
Z
u
0
1
2
e
u
du
=1
e
u
=1
e
x
2
F
(
x
)=
Ëâ€
1
e
x
2
for
x>
0
0
for
x
0
1
Question3
Considerthefollowingfunction
f
(
x
)=
x
2
+6
x
40
;
for
x
=1
;
2
;
3
;
4
a)Verifythatitcanserveasthecumulativedistributionfunctionofadiscreterandomvariable
F
(
x
)=
8
>
>
>
>
<
>
>
>
>
:
0
for
x<
1
7
40
for
1
x<
2
16
40
for
2
x<
3
27
40
for
3
x<
4
1
for
x
4
b)ËÂndtheprobabilitydistributionof
X
f
(
x
)=
8
>
>
<
>
>
:
7
40
for
x
=1
9
40
for
x
=2
11
40
for
x
=3
13
40
for
x
=4
c)ËÂnd
P
(1
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