http://www.math.ilstu.edu/krzysio/
Krzys’ Ostaszewski
Author of the BTDT Manual for Course P/1
available at http://smartURL.it/krzysioP or http://smartURL.it/krzysioPe
Instructor for online Course P/1 seminar: http://smartURL.it/onlineactuary
Exercise for September 24, 2005
Let X1 and X2 be a random sample from the uniform distribution on [0, 1] and let
Y1 = min ( X1 , X2 ) , Y2 = max ( X1 , X2 ) . Find fY1 ( y1 Y2 = y2 ) .
A.
1
2
B.
1
2y2
C.
1
y2
D. 2
E. 1
Solution.
Recalling the formula for the joint PDF of two order statistics, we obtain immediately:
fY(1) ,Y(2) ( y1 , y2 ) = 2!! f X ( y1 ) ! f X ( y2 ) = 2!!1!1 = 2.
This actually tells us that the joint PDF of the two order statistics is uniform on the region
{0 ! y1 ! y2 ! 1}. We conclude then that the conditional distribution of Y1 Y2 = y2 is
uniform on [ 0, y2 ] by looking at the figure below
y2
Area where y1 ! y2
1
Line y2 = y1
y2
1
y1
Y1 Y2 = y2 is distributed on this line,
and the distribution is uniform,
because the joint distribution is
uniform on the shaded triangle.
Answer C.
© Copyright 2005 by Krzysztof Ostaszewski.
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