Unsupervised Learning  Homework on Sufficiency 

Directions: Be sure to show your work and explain your answer for each question. Your 
homework solutions are to be entirely your own effort. You may not communicate with anyone 
about the homework, except for the TA and/or the instructor. You may use the Canvas postings, 
in-class discussion, and any of the recommended textbooks (if appropriate). 
Note on submission: Please submit your solutions as a single Word file. Word is editable and 
will facilitate TA grading comments. You do not need to use an equation editor for math text. 
You may hand-write solutions. Scan all hand-written solutions (be sure they are legible) and 
embed them in a single Word file. 
1. [mazon fulfillment centers want to ensure a uniform (and low) processing time 
for orders. At one center, Amazon tracked a random sample of n orders and compared the actual 
processing time of each order against Amazon’s standard. The amount of time x that an order 
departed early was recorded with a negative sign (x < 0) or late with a positive sign (x > 0). For 
the analysis, the following statistical model was used for the x’s: Suppose that 
are 
independent random variables with common density function 
, for 
, i = 1, 2, …, n, where 
is an unknown parameter. A small value for 
represents uniformity of processing times. Find a one-dimensional sufficient statistic for . 
2.  Computers make small “machine” errors in floating point operations that can 
accumulate across complex calculations. As a test, a new computer chip was given a series of n 
complex calculations for which the answers were known. For each calculation, i = 1, 2, …, n, the 
machine error was recorded. Interest focuses upon the distribution of machine errors (mean, 
variance, maximum error, etc.) The following statistical model was adopted for the machine 
errors: Suppose that 
are independent random variables with common density 
function 
1 
for 
( ; ) 
2 
0 otherwise 
i 
i 
x 
f x 
  
 
 
 
−   + 
= 

 
, i = 1, 2, …, n, where 
is an unknown 
parameter. Find a one-dimensional sufficient statistic for  and hence for the questions of 
interest. [Hint: Note the limitations on the range of X.] 
X1, X2 ,...,Xn 
−   xi   
  0 
 
xi 
X1, X2 ,...,Xn 
  0 
2
2 
( ; ) 1 
 
   
i 
x 
f xi e 
− 
=3. [Suppose that 
are independent random variables with common 
density function 
, for 
, i = 1, 2, …, n, where 
are unknown parameters. Let 
be the ordered values of 
. That is, 
are 
rearranged in order so that 
. 
Specifically, 
. Show that 
are 
sufficient statistics for 
. 
[Hint: This problem can be solved easily by using either the definition of sufficiency or the 
Factorization Theorem when thought about in the right way. To use the definition, for example, 
suppose n=3 and 
. Then what is the conditional probability that 
given that 
? That is, if you know that your data are the 
values 1, 2, 3, what is the probability that they occurred in the sequence 3, 1, 2?] 
X1, X2 ,...,Xn 
2 
)2 
( 
( ; , ) 12 

 
    
− 
− 
= 
i 
x 
f xi e 
−   xi   
−     ,  0 
Y1,Y2 ,...,Yn 
X1, X2 ,...,Xn 
Y1,Y2 ,...,Yn 
X1, X2 ,...,Xn 
Y1  Y2  Yn 
Y1,Y2 ,...,Yn 
, 
y1 =1, y2 = 2, y3 = 3 
x1 = 3, x2 =1, x3 = 2 
y1 =1, y2 = 2, y3 = 3 
Y1 = min( X1, X2 ,...,Xn ),...,Yn = max(X1, X2 ,...,Xn )