Mathematical Statistics (Math 436) – Exam #1
Spring 2006 – Professor Hartlaub
Name
This exam contains two parts, a take home part and an in class part. The point values for each
problem are provided in parentheses. Good luck and have a great break!
Part 1 – Take Home Problems. Please remember the honor code that we discussed in class.
You may not discuss these problems with anyone else and you should report any violations of
the honor code that you observe.
1. Let Y1 , Y2 ,
, Yn be a random sample from a population with density function
 3y2
,0  y  

f ( y | )    3
 0, elsewhere.
We have previously showed that Y( n )  max Y1,Y2 ,
,Yn  is sufficient for  .
a. Find the MLE for  . (5)
b. Find a function of the MLE in part (a) that is a pivotal quantity. (10)
c. Use the pivotal quantity from part (b) to find a 100(1   )% confidence interval for  .
(10)
2. Suppose that Y1 , Y2 ,
, Yn constitute a random sample of size n from an exponential
distribution with mean  . Find a 100(1   )% confidence interval for t     2 . (15)
Part 2 – In Class Problems. Please show your work for all of the problems below.
3. Let Y1 , Y2 ,
, Yn be independent random variables such that each Yi has a gamma distribution
n
with parameters i and  . Prove that U   Yi has a gamma distribution and identify the
i 1
parameters. (15)
4. Let Y1 , Y2 ,
, Yn be a independent random variables, each with probability density function
3 y 2 , 0  y  1
f ( y)  
 0, elsewhere.
Show that Y converges in probability to some constant, and find the constant. (15)
5. Let Y1 , Y2 ,
, Y10 denote a random sample of size ten from a normal distribution with mean 0
and variance  2 .
a. Identify the distribution of
Y
. (5)
 S 


 10 
2
10Y
. (5)
S2
S2
c. Identify the distribution of
. (5)
2
10Y
b. Identify the distribution of
S


d. Use your answer from part (c) to find the number c such that P  c   c   0.95. (5)
Y


6. Let Y1 , Y2 ,
, Yn be independent and identically distributed uniform random variables on the
interval  0,3  .
a. Derive the method of moments estimator for  . (5)
b. Find the maximum likelihood estimator for  . (5)
7. Suppose that Y1 , Y2 , , Yn denote a random sample from a population with probability density
function
 y 
 1


e    2  , y  0,   2

f ( y)     2

0, elsewhere.

a. Find a sufficient statistic for  . (5)
b. Suggest a statistic to use as an unbiased estimator of  ? (5)
c. Is the unbiased estimator you suggested in part (b) an MVUE for  ? Explain why or
why not. (10)
d. Derive the MSE of your estimator in part (b). (5)