ONLY 3 PROBLEM probability of normal continuous distributions

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Gvat8

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only 3 question of probability of normal continuous distributions. please step by step. all the requirement are on the pdf.

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MTH 34200 – HW 06 (20 points) – Due Tuesday, 10/9 Name_______________________________________ 1. (4 pts) In each of the four plots below, the red triangle displays the value of a parameter 𝜃. The blue dots in ̂1 , 𝜃 ̂2 , 𝜃 ̂3 , and 𝜃 ̂4 (not necessarily in that order). each plot represent observations of one of four estimators 𝜃 a) b) c) d) The bias and variance of each of the four estimators is provided below. For each estimator, calculate the MSE and identify which plot above is most likely to show observations of that estimator. ̂1 ] = 0 𝐵[𝜃 ̂1 ] = 9 Var[𝜃 ̂1 ] = ____________ 𝑀𝑆𝐸[𝜃 Plot: ____________ ̂2 ] = −2 𝐵[𝜃 ̂2 ] = 9 Var[𝜃 ̂2 ] = ____________ 𝑀𝑆𝐸[𝜃 Plot: ____________ ̂3 ] = −1 𝐵[𝜃 ̂3 ] = 4 Var[𝜃 ̂3 ] = ____________ 𝑀𝑆𝐸[𝜃 Plot: ____________ ̂4 ] = 4 𝐵[𝜃 ̂4 ] = 3 Var[𝜃 ̂4 ] = ____________ 𝑀𝑆𝐸[𝜃 Plot: ____________ 2 2. (6 pts) Let 𝜃̂ be an estimator for a parameter 𝜃. Show that 𝑀𝑆𝐸[𝜃̂] = Var[𝜃̂] + (𝐵[𝜃̂]) . ̂1 be an estimator for a parameter 𝜃. Assume that 𝐸[𝜃 ̂1 ] = 𝑘𝜃 and Var[𝜃 ̂1 ] = 𝑐𝜃 2. 3. (10 pts) Let 𝜃 ̂1 ] in terms of 𝜃, 𝑐, and 𝑘. a) Find 𝐵[𝜃 ̂1 ] in terms of 𝜃, 𝑐, and 𝑘. b) Find 𝑀𝑆𝐸[𝜃 ̂2 for 𝜃. Define 𝜃 ̂2 as a function of of 𝜃 ̂1 . c) Define a new unbiased estimator 𝜃 ̂2 ] in terms of 𝜃, 𝑐, and 𝑘. d) Find Var[𝜃 ̂2 ] in terms of 𝜃, 𝑐, and 𝑘. e) Find 𝑀𝑆𝐸[𝜃 f) ̂2 ] < 𝑀𝑆𝐸[𝜃 ̂1 ]. Find a value of 𝑘 for which 𝑀𝑆𝐸[𝜃 ̂1 ] < 𝑀𝑆𝐸[𝜃 ̂2 ]. g) Find values of 𝑐 and 𝑘 for which 𝑀𝑆𝐸[𝜃
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MTH 34200 – HW 06 (20 points) – Due Tuesday, 10/9

Name_______________________________________

̂1 be an estimator for a parameter 𝜃. Assume that 𝐸[𝜃
̂1 ] = 𝑘𝜃 and Var[𝜃
̂1 ] = 𝑐𝜃 2.
3. (10 pts) Let 𝜃
̂1 ] in terms of 𝜃, 𝑐, and 𝑘.
a) Find 𝐵[𝜃
Bias=sum of variance/no of periods
̂1 ] = ⅀𝑐𝜃 2 /( 𝐸[𝜃
̂1 ] = 𝑘𝜃)
Bias=sum of Var[𝜃
= ⅀𝑐𝜃 2 / 𝑘𝜃
̂1 ] in terms of 𝜃, 𝑐, and 𝑘.
b) Find 𝑀𝑆𝐸[𝜃

MSE=Sum of Variance/n-1
̂1 ] − 1 = 𝑘𝜃-1)
𝑀𝑆𝐸 = ⅀𝑐𝜃 2/( 𝐸[𝜃
= ⅀𝑐𝜃 2 /( 𝑘𝜃-1)
̂2 for 𝜃. Define 𝜃
̂2 as a function of of 𝜃
̂1 .
c) Define a new unbiased estimator 𝜃
̂2 =( 𝑘𝜃
̂1 + 1)/𝑘𝜃
̂1
𝜃

̂2 ] in terms of 𝜃, 𝑐, and 𝑘.
d) Find Var[𝜃
̂2 ] = 𝑐𝜃 2 *(k 𝜃) /(k �...


Anonymous
Excellent resource! Really helped me get the gist of things.

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