Suppose that X1,...,X5 is a random sample from the uniform distribution with parameter θ (i.e., these are independent uniform random variables with parameter θ, where θ is unknown). Define the following estimator (a random variable that is a function of X1,...,X5): Xmax = k max(X1,...,X5), where k will be selected later. (a) Find the density of Xmax (it will depend on θ and k). (b) Find k such that Xmax is an unbiased estimator for θ.

Suppose that X1,...,X5 is a random sample from the uniform distribution with parameter θ (i.e., these are independent uniform random variables with parameter θ, where θ is unknown). Define the following estimator (a random variable that is a function of X1,...,X5): Xmax = k max(X1,...,X5), where k will be selected later. (a) Find the density of Xmax (it will depend on θ and k). (b) Find k such that Xmax is an unbiased estimator for θ.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter13: Probability And Calculus

Section13.CR: Chapter 13 Review

Problem 1CR

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Question

Suppose that X1,...,X5 is a random sample from the uniform distribution with parameter θ (i.e., these are independent uniform random variables with parameter θ, where
θ is unknown). Define the following estimator (a random variable that is a function of X1,...,X5): Xmax = k max(X1,...,X5), where k will be selected later.

(a) Find the density of Xmax (it will depend on θ and k).
(b) Find k such that Xmax is an unbiased estimator for θ.

Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.

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