Stat 415 Exam #1
Student Name: Date:
You have 105 minutes to complete and upload. You must show all of your
work in order to receive full credit. The use of software is not permitted
on this exam. 7 pages, 100 points
1. 25 points Suppose we have a distribution with parameter θ > 0 defined by the
density function:
2
f (y) = 2yθe−θy , y>0
and zero otherwise. Let Y1 , Y2 , . . . , Yn be a random sample from this distribu-
tion.
(a) 15 points Find the MLE of θ.
, (b) 5 points Find the sufficient statistic for θ. To receive full credit, provide all
details including which theorem was used and all functions clearly defined.
(c) 5 points Is the MLE a sufficient statistic? Justify.
Student Name: Date:
You have 105 minutes to complete and upload. You must show all of your
work in order to receive full credit. The use of software is not permitted
on this exam. 7 pages, 100 points
1. 25 points Suppose we have a distribution with parameter θ > 0 defined by the
density function:
2
f (y) = 2yθe−θy , y>0
and zero otherwise. Let Y1 , Y2 , . . . , Yn be a random sample from this distribu-
tion.
(a) 15 points Find the MLE of θ.
, (b) 5 points Find the sufficient statistic for θ. To receive full credit, provide all
details including which theorem was used and all functions clearly defined.
(c) 5 points Is the MLE a sufficient statistic? Justify.
