410 STAT Spring 2008 Version B
Name
(10 points)
ANSWERS
.
Quiz 1
Be sure to show all your work, your partial credit might depend on it.
No credit will be given without supporting work.
1.
Consider a continuous random variable X with p.d.f.
fX( x) =
2 x 21 0
2< x<5 otherwise
a) (4) Find the moment-generating function M X ( t ).
5 2
M X ( t ) = E( e t X ) =
=
=
etx
2 2 5 xdx = xet x d x 21 21 2
5 2
2 1 1 xet x - 2 et x 21 t t
10 2 4 2 e 5t - e 5t - e 2t + e 2t , 2 2 21 t 21 t 21 t 21 t
t 0.
M X ( 0 ) = 1.
b) (6) Let Y =
1 X2
. Find the probability distribution of Y.
( That is, either find the c.d.f. of Y, F Y ( y ), y < < ,
OR find the p.d.f. of Y, f Y ( y ), < y < . )
fX( x) =
2 x 21 0
0
x<2
2 x<5
2< x<5
FX( x ) =
x2 -4
21 1
otherwise
x5
Y=
1 X2
.
2<x<5
1 1 <y< . 25 4
1
1/2
g( x) =
1
x2
g 1( y ) =
y
=y
dx 1 3 = y /2 dy 2
1 1 <y< . 25 4
f Y ( y ) = f X ( g 1( y ) )
2 1 /2 1 3 /2 1 2 dx =( y )( y )= y , 21 2 21 dy
OR 1 X
2
FY( y ) = P( Y y ) = P( = 1 FX(
y ) = P( X
1
y
) = 1 P( X <
1
y
)
1
y
). = 1
y -1 - 4
21
25 - y -1 = , 21
1 1 <y< . 25 4
0
FY( y ) =
y<
1 25
25 - y -1 21 1
1 1 y< 25 4
fY( y) =
1 -2 y 21 0
1 1 <x< 25 4 otherwise
y
1 4