[SOLVED] STA280-Homework 1

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 X₁, X₂,…,X _n are

1 −_x₂i2 independent random variables with common density function f (x_i ;) =          e          , for



−x_i , i = 1, 2, …, n, where 0is 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 x_i 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 X₁, X₂,…,X _n are independent random variables with common density function f x( _i;) =^_^2¹ ^{for − xi +}, i = 1, 2, …, n, where 0is an unknown

_ 0         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.]

3. Suppose that X₁, X₂,…,X _n are independent random variables with common

−(x_i −)²

density function f (x_i;,) = , for −x_i , i = 1, 2, …, n, where

−, 0are unknown parameters. Let Y₁,Y₂,…,Y_n be the ordered values of

X₁, X₂,…,X _n . That is, Y₁,Y₂,…,Y_n are X₁, X₂,…,X _n rearranged in order so that Y₁ Y₂ LY_n. Specifically, Y₁ = min(X₁, X₂,…,X _n ),…,Y_n = max(X₁, X₂,…,X _n ). Show that Y₁,Y₂,…,Y_n 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 = 2given 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?]