HW 8
Due: Mar 29, 2021. 23:59 pm.
Scan or take photos of your written homework, combine them into a single pdf, and upload on ICON.
Textbook 5.2.1, 5.2.2, 5.2.3, 5.2.9
1. Let X1, · · · , Xn be (mutually) independent random variables, each follows the Uniform distribution on (−θ, θ). Here, the parameter θ is an (unknown) positive number.
(a) Write down the pdf of X1 and sketch a plot of it.
(b) Derive the cdf of X1 and sketch a plot of it.
(c) Let Tn = max{X1,··· ,Xn}. Derive the cdf and pdf of Tn.
(d) Show that Tn is a consistent estimator of θ. That is, show that Tn converges in probability to θ as
n → ∞.
2. Assume that
Yi =2+βi+εi, i=1,2,…,n.
where ε1, . . . εn is a random sample from normal distribution with zero mean and variance σ2. Let
ˆ
(a) Show that βˆ is an unbiased estimator for β.
ni=1(Yi − 2)i β= ni=1i2 .
(b) Show that βˆ converges in probability to β as n → ∞.
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