Let (x1, x2,,xn) be i.i.d. samples from a distribution with mean μ and variance o2. Consider the following estimator of μ: μ* = Note that x = n(n + 1) (2n + 1)/6. n Σi=1 ixi Σ11 (a) Show that μ* is an unbiased estimator of u. (b) Explain why the above estimator * is less preferable compared to using the sample average as an estimator for μ. Hint: Show that * has a larger mean squared error. (1/n) (Σ=1*i), Σ = n(n+1)/2 and Σ1 i² :1
Let (x1, x2,,xn) be i.i.d. samples from a distribution with mean μ and variance o2. Consider the following estimator of μ: μ* = Note that x = n(n + 1) (2n + 1)/6. n Σi=1 ixi Σ11 (a) Show that μ* is an unbiased estimator of u. (b) Explain why the above estimator * is less preferable compared to using the sample average as an estimator for μ. Hint: Show that * has a larger mean squared error. (1/n) (Σ=1*i), Σ = n(n+1)/2 and Σ1 i² :1
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![Let (x1,x2,...,xn) be i.i.d. samples from a distribution with
mean μ and variance o². Consider the following estimator of µ:
леж
i=1
Σ₁=₁ ixi
Σ11
(a) Show that µ* is an unbiased estimator of μ.
(b) Explain why the above estimator u* is less preferable compared to
using the sample average à as an estimator for μ.
Hint: Show that * has a larger mean squared error.
Note that x
n(n+1)(2n +1)/6.
(1/n) (Σï=1 xi), Σ₁_₁ i n(n+1)/2 and Σ₁²
i=1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea47a4b1-a227-4b18-bcf9-7f7b72a7a749%2F424ed472-0cb2-4199-b975-1942d3a13ed9%2Fg3l2k5j_processed.png&w=3840&q=75)
Transcribed Image Text:Let (x1,x2,...,xn) be i.i.d. samples from a distribution with
mean μ and variance o². Consider the following estimator of µ:
леж
i=1
Σ₁=₁ ixi
Σ11
(a) Show that µ* is an unbiased estimator of μ.
(b) Explain why the above estimator u* is less preferable compared to
using the sample average à as an estimator for μ.
Hint: Show that * has a larger mean squared error.
Note that x
n(n+1)(2n +1)/6.
(1/n) (Σï=1 xi), Σ₁_₁ i n(n+1)/2 and Σ₁²
i=1
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