tugas teopel1

Task 2
Teori Peluang
Nama (NIM) :
1. Nur Azlindah
2. Novia Ani Sa’ada
3. Ika Nur Khasana
4. Ririn Wulan Mei

(14610005)
(14610018)
(14610022)
(14610028)

Continuous Random Variables
Problem 1
Choose a real number uniformly at random in the interval (2,6) and call it X
a. Find the CDF of X, Fx(X).
b. Find EX
Answer :
a. CDF of X, Fx X
=

=

=

−

−

=∫
=

dx

= −

=

Fx X {

−

−

|

,

<

>

b. Expected Value (EX)
∞
= ∫−∞ X F x dx
=∫

=∫
=

=

=

EX =

+

−

=

+

Problem 4

=

=4

Let X be a uniform (0,1) random variable, and let
a. Find the CDF of Y
b. Find the PDF of Y
c. Find EY.
Answer:
� =
� =

,

−

,

a. The CDF of Y
� = − ,
=�
For < − ,
=�
For

,
=�
=� − .
=� −
ln
=�
−ln
∞

= ∫

− ln
∞

= ∫

− ln
∞

= ∫

− ln

−

=

=

=

− .

∞
= |−
ln

=
=

− − ln

+ ln

−

,

<

So, The CDF of Y is

={
b. The PDF of Y

,

,

<

−

−

<

]

[

=

+ ln

,

[ + ln ]

=

=

So, The PDF of Y is
={
∞

,

,

−

= ∫−∞ ∙

c.

= ∫
�−

<

ℎ

�

∙

= ∫
�−

= |� −
= −

−

Problem 6
Let
that

∼

∼

� �

� �

, and
.

=

, where is a positive real number. Show

Answer:
CDF of Y
=�
=�

=�

=

So, CDF of Y is
=

=

[

=

=−

=−

=

−

−

]

−

[ −
−

∙

�
�

]

[−

−
∙( )

−
−

=

−

−

( )

∼

So, it is evident that
Problem 8
Let x ~ N (3,9)
a. Find P (x > 0)
b. Find P (-3 < x < 8)
c. Find P (x > 5|x > 3)

Answer :
X is a normal random variable with
a. P(x > 0) = FX (0)
= Φ

–

= Φ −
=Φ
= 0.46

. From this, we can write the PDF of Y

]

� �

=

.

dan σ = √ = , thus we have

b. P(-33) =

=

P x> , x>

=

P x>
P x>
P x>
–Φ
–Φ
−Φ
–Φ

=

=

Problem 9
Let

.

–
.

>
We have
Thus
�

>

=

=
=

=

−Φ
.

.

– .

~� ,
and =
a. Find �
>
b. Find � − < <
c. Find � > | <

Answer :
a. �

−

–Φ

=

=

5−

= 0.5

−

and �

−Φ

−Φ

�

→ �=

=

−

−

− Φ −

= − ,
= ,

b. Find � − < <
Using theorem
= −

−

=

. Since

,

~�
,�
ℎ
=
+
� =
� = −
So ~ � ,
Therefore
� − <

<

= −
=

+

=

−
−
)−Φ
�
�
−
− −
= Φ(
) − Φ(
)

= Φ(

= Φ( ) − Φ −

c. Find �
�

> | <

> | <

= ,
= ,

− ,

=�

> | −

> | >

=�

=
=

�

�
�

=
=
=
=

= ,

<

> , >
� >
>
>

− Φ
−Φ

− Φ
−Φ

− Φ

,

−Φ

,

−
�
−
�

−

−

Problem 11
Let x~Exponential (2) and Y=2+3x
a. Find P(x>2)
b. Find E[Y] and Var(Y)
c. Find P(x>2|Y
�
>
=
=
=

b.

= −
= −
= −

=

=

�

c. �

=

=�

=

+ x
+
+

+
=

+

+

+
+�
( )

=

( )=

>

|

<

=�
=

=

=

�

�

�

−

>

| +

<

<

> , <
� <
<

− −
− −

>

Problem 14
Let X be a random variable with the following CDF

;

=
{

+

<

;

;

<

<

;

a. Find the generalized PDF of X,
b. Find
using
c. Find �
using
.
Answer:

The graph CDF of X

FX(x)
1,5
1
0,5
0
0

0,5

< ,

For
=

=

=

[

[ ]

]

1

1,5 x

For

< ,
=

]

[

=

=

[ + ]

From the graph PDF of X.

The graph PDF of X

fx(x)
1,2
1
0,8
0,6
0,4
0,2
0
0

0,5

a. The PDF of X,
=

=

=
b.

−

−

=

using
=

=

=

+

+

,
,

1

1,5

x

c. �

=

using
�

=
=

=
=

=

−

−