The Islamic University of Gaza
Faculty of Commerce
Economics & Applied Statistics Department
Mathematical stat. – Instructor : Ibrahem Abed
Dec.. 29th, 2009 Time: 1 hour
Second Mid. Exam
Question #1:( 6 points)
Suppose that Y1 , Y2 ,..............., Yn constitute a random sample from the density function
e ( y  )
f (y |)  
 o
y 
,
,
elsewhere
Where  is an unknown positive constant

Find an estimator  1 for  by using the method of moments.
Question #2:( 12 points)
Suppose that Y1 , Y2 ,..............., Yn denote a random sample from the poisson distribution with mean 

a) Find the MLE estimator  for 

b) Find the Expected value and variance of 
c) Show that the estimator of (a) consistent for 
d) What is the MLE for P(Y  0)  e 
Question #3:( 20 points)
Suppose that Y1 ,Y2 denote a random sample from an exponential distribution with density function
 1  y
,
y0
 e
f ( y )  
 o
, elsewhere
Consider the following estimator of 


1  Y ,  2 
1
Y1  Y2
,
2

3  Y
1
 2Y2
3

,
4  Y
___
a) Which of these estimators are unbiased ?
b) Among the unbiased estimators , which has the smallest variance ?

c) Find the efficiency of
3

relative to
2

and
1
1
respectively .
Question #4 :( 15 points)
Suppose that Y1 , Y2 ,..............., Yn denote a random sample from Weibull density distribution ,given by
 2 y  y2 
,
 e
f (y | )   
 o
,
y0
elsewhere
Find a MVUE for 
Question #5:( 7 points)
If Y1 , Y2 ,..............., Yn denote a random sample from a geometric distribution with parameter p
__
Show that Y is sufficient for p
Good luck ☺☺☺
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