P_PracticeExam6_05-1..
P_PracticeExam6_05-1..
P_PracticeExam6_05-1..
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with density function<br />
! 2x, for 0 < x < 1,<br />
fX ( x)<br />
= "<br />
# 0, otherwise.<br />
What is E( X( 6)<br />
)?<br />
A. 1<br />
2<br />
2<br />
B.<br />
3<br />
5<br />
C.<br />
6<br />
Solution.<br />
The CDF of the original distribution is:<br />
x<br />
FX ( x)<br />
= ! 2tdt = t 2<br />
0<br />
t = x<br />
t =0<br />
= x 2<br />
6<br />
D.<br />
7<br />
12<br />
E.<br />
13<br />
PRACTICE EXAMINATION NO. 6<br />
for 0 < x < <strong>1.</strong> We also have FX ( x)<br />
= 0 for x ! 0, and FX ( x)<br />
= 1 for x ! <strong>1.</strong> Therefore,<br />
FX ( x)<br />
= Pr X<br />
( 6)<br />
( ( 6)<br />
! x)<br />
= ( FX ( x)<br />
) 6<br />
= x 12<br />
for 0 < x < <strong>1.</strong> Also, FX ( x)<br />
= 0 for x ! 0 and F<br />
( 6)<br />
X ( x)<br />
= 1 for x > <strong>1.</strong> Hence, for 0 < x < 1,<br />
( 6)<br />
sX ( x)<br />
= 1 ! x<br />
( 6)<br />
12 ,<br />
( x)<br />
= 0 for x ! <strong>1.</strong> Therefore,<br />
and sX( 6)<br />
Answer E.<br />
( ) = 1 ! x 12<br />
1<br />
" ( )dx = x ! x13 # &<br />
$<br />
%<br />
13 '<br />
E X 6<br />
( )<br />
0<br />
1<br />
( 0<br />
= 12<br />
13 .<br />
6. February 1996 Course 110 Examination, Problem No. 2<br />
Let X be a normal random variable with mean 0 and variance a > 0. Calculate Pr X 2 ( < a).<br />
A. 0.34 B. 0.42 C. 0.68 D. 0.84 E. 0.90<br />
Solution.<br />
Since a > 0, and a is the variance, a is the standard deviation of X. Therefore, if we denote a<br />
standard normal random variable by Z, and the standard normal CDF by !, we get<br />
Answer C.<br />
( ) = Pr<br />
! a ! 0<br />
<<br />
a<br />
X ! 0<br />
"<br />
a ! 0%<br />
$<br />
<<br />
#<br />
a a<br />
'<br />
&<br />
=<br />
= Pr ( !1 < Z < 1)<br />
= ( ( 1)<br />
! ( ( !1)<br />
= ( ( 1)<br />
! ( 1! ( ( 1)<br />
) =<br />
= 2( ( 1)<br />
! 1 = 2 ) 0.84134474 ! 1 = 0.68268948.<br />
Pr X 2 ( < a)<br />
= Pr ! a < X < a<br />
ASM Study Manual for Course P/1 Actuarial Examination. © Copyright 2004-2007 by Krzysztof Ostaszewski - 261 -
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