Description
1. Amazon 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 X 1 , X 2 ,…, X n are
−xi2
2
1 independent random variables with common density function f ( xi ; ) = e
, for − xi , i = 1, 2, …, n, where 0 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 xi 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 X 1 , X 2 ,…, X n are independent random variables with common density
for − x +
i , i = 1, 2, …, n, where 0 is an unknown
otherwise
parameter. Find a one-dimensional sufficient statistic for and hence for the questions of interest. [Hint: Note the limitations on the range of X.]
1 function f ( x ; ) = 2
i 0
3. [10 points] Suppose that X 1 , X 2 ,…, X n are independent random variables with common
−(xi −)2
2
1 density function f (xi ;, ) = 2 e
, for − xi , i = 1, 2, …, n, where
− , 0 are unknown parameters. Let Y1 ,Y2 ,…,Yn be the ordered values of
X1,X2,…,Xn . That is, Y1,Y2,…,Yn are X1,X2,…,Xn rearranged in order so that Y1 Y2 Yn . Specifically, Y1 = min( X 1 , X 2 ,…, X n ),…,Yn = max( X 1 , X 2 ,…, X n ) . Show that Y1 ,Y2 ,…,Yn 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 y1 =1, y2 = 2, y3 = 3. Then what is the conditional probability that
x1 = 3, x2 = 1, x3 = 2 given that y1 = 1, y2 = 2, y3 = 3? 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?]
