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Content type
User Generated
Subject
Probability
School
École polytechnique fédérale de Lausanne
Type
Homework
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The density function of
~ Uniform(2 )T
is
1
()
2
T
ft
=
whenever
[0;2 ]t
.
a)
2
0
2
22
0
1 1 (2 )
E( ) (
22
)
2 4
T
t dt dt
t
T tf t
+
−
= = = = =

;
b)
0
2
0
2
( ) sin( )
11
E( ) ( ) cos( ) 0
22
T
t dt t dtU u t f t

+
−
= = = − =

;
c)
2
2
2
0
2
00
1 cos(2 )
( ) sin ( )
2 4 4
1 1 1 sin2 2 1
E( ) ( )
2 2 2 2
T
t
t dt t dt dpf t
t
P t t

+
−
−

= = = = − = =


.

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The density function of T ~ Uniform(2 ) is fT (t ) = + 1 a) E(T ) =  tfT (t )dt = 2 − 2 + b) E(U ) =  u(t ) f − + T (t )dt = 1 t2 t dt = 0 2 2 1 2 1 c) E( P) =  p(t ) fT (t )dt = 2 − 2 2 = 0 1 whenever t  [0; 2 ] . 2 ...
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