P_PracticeExam6_05-1..

20
! z
10
PRACTICE EXAMINATION NO. 6
This case ends when the line crosses the origin, i.e., when z = 0. Between z = 0 and z = 300, the
probability we want to find is just the area of the shaded region as a fraction of the area of the 20
by 20 square. The bottom side of the shaded region has length z
(from y = 0 and the equation
25
of the line) and the top side has length 8 + z
(from y = 20 and the equation of the line), so that
25
its area is 20 ! 4 + z " %
#
$
25&
' = 80 + 4
z. As a fraction of the entire area, this is
5
Therefore,
80 +
FZ ( z)
=
4
5 z
400
= 1 z
+
5 500 .
fZ ( z)
= FZ! ( z)
= 1
for 0 ! z ! 300.
500
Again, this is the same answer we obtained before. The final case is when the line crosses the yaxis
above the origin, but below 20. When the line crosses the origin, z = 0. When the line
crosses the y-axis at the point (0, 20), we have z = !200. In this case, the figure looks as follows:
! z
10
y
y = 2.5x ! 0.1z
y = 2.5x ! 0.1z
ASM Study Manual for Course P/1 Actuarial Examination. © Copyright 2004-2007 by Krzysztof Ostaszewski - 269 -
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