Task 2
Teori Peluang Nama (NIM) :
1. Nur Azlindah (14610005) 2. Novia Ani Sa’ada (14610018) 3. Ika Nur Khasana (14610022) 4. Ririn Wulan Mei (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 {
<
−
>
= |− ln∞
= − − ln
= + ln , − <
So, The CDF of Y is
= { , <
−
+ ln , − <
,
b. The PDF of Y
= [ ]
= [ + ln ]
=
So, The PDF of Y is
= { , − <
, ℎ �
c. = ∫−∞∞ ∙
= ∫ ∙
�−
= ∫
�−
= |�−
= − −
Problem 6
Let ∼ � � , and = , where is a positive real number. Show
b. P(-3<x<8) = FX (8) - FX (-3) Using theorem
~ � , �
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<11) Answer :
a. � > = −
� > = −
= −
b. = + x
= +
= +
= +
= +
=
� = +
= � + �
= + ( )
= ( ) =
c. � > | < = � > | + >
=� > , <� <
= � < <� <
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:
For < ,
= [ ]
= [ ]
= 0 0,5 1 1,5
0 0,5 1 1,5
FX(x)
x
For < ,
= [ ]
= [ + ]
=
From the graph PDF of X.
a. The PDF of X,
= − ,
= − ,
=
=
b. using
= +
= +
= 0 0,2 0,4 0,6 0,8 1 1,2
0 0,5 1 1,5
fx(x)
x
=
c. � using
� = −
= −
=
=
