Math 324 A2 v6.pdf - Alex X. Liu 260867551 Math 324...

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Alex X. Liu 260867551 Math 324 Assignment 2 1. 𝐿?? ? = ?̅− ?̅?ℎ??,? ~ 𝑁 (0,𝜎2?+3𝜎2?)? ~ 𝑁 (0,4𝜎2?)??? ?? ???????? ?????? ????,𝑃 (|?̅− ?̅| < 𝜎5) = 𝑃 (|?| <𝜎5) = 2𝑃 (0 < ? <𝜎5) 𝐿?? ? =(? − 0)4𝜎2?= ?√?2𝜎~𝑁(0,1)2𝑃 (0 < ? <𝜎5) = 2𝑃 (0 <?√?2𝜎<𝜎√?10𝜎) = 2𝑃 (0 < ? <√?10)?ℎ??,2𝑃 (0 < ? <√?10) = 0.90𝑃 (0 < ? <√?10) = 0.45????????? ?? ?????,√?10≈ 1.64? ≈ 16.42≈ 269
2. a) ? ~ ?????????(0.5)𝜇 = 0.5, 𝜎2= 0.25????? ? ?? ????? (> 30), ?? ?ℎ? ?𝐿?, ?𝑛≈ 𝑁(?𝜇, ?𝜎2)𝑃(?𝑛≥ 5100) = 𝑃 (?𝑛− 10000 ∗ 0.5√10000∗ 0.5>5100 − 10000 ∗ 0.5√10000∗ 0.5) = 𝑃(? > 2 )?ℎ??? ? ~𝑁(0,1)𝑃(? > 2) = 1 − 𝑃(? < 2) = 1 − 0.9772 =0.0228 ≈ 𝑃(?𝑛≥ 5100)b) ?? ?ℎ? ?𝐿?, ?𝑛≈ 𝑁 (𝜇,𝜎2?)𝑃(40% < ?𝑛< 60%) = 𝑃 (0.4 − 0.50.5√?<?𝑛− 0.50.5√?<0.6 − 0.50.5√?)≈ ? (−0.1√?0.5< ? <0.1√?0.5) = 2𝑃 (0 < ? <15√?)?ℎ??? ? ~𝑁(0,1)2𝑃 (0 < ? <15√?) = 0.95𝑃 (0 < ? <15√?) = 0.475????????? ?? ?????,15√?= 1.96√?= 5 ∗ 1.96
? = (5 ∗ 1.96)2? = 96.04 ≈ 963. a) MME ?(?) = ∫???(?)??−∞= ∫?(𝜃 + 1)?−𝜃−2??1= (𝜃 + 1) ∫? ∗ ?−𝜃−2??1= (𝜃 + 1) ∫?−𝜃−1??1= (𝜃 + 1) [−?−𝜃𝜃 ]1=(𝜃 + 1)𝜃= 1 + 1/𝜃𝐿?? ?̅= 1 + 1/𝜃𝜽̂??𝑬=??̅− ?MLE ℒ(𝜃) = ∏ ??(?𝑖)𝑛𝑖=1= ∏(𝜃 + 1)?𝑖−𝜃−2𝑛𝑖=1= (𝜃 + 1)𝑛∏ ?𝑖−𝜃−2𝑛𝑖=1
ℓ(𝜃) = ????(𝜃 + 1) + ∑ log(?𝑖−𝜃−2)𝑛𝑖=1ℓ(𝜃) = ????(𝜃 + 1) + ∑(−𝜃 − 2) log(?𝑖)𝑛𝑖=1ℓ(𝜃) = ????(𝜃 + 1) + (−𝜃 − 2) ∑ log(?𝑖)𝑛𝑖=1ℓ(𝜃) = ????(𝜃 + 1) − 𝜃 ∑ log(?𝑖)𝑛𝑖=1− 2 ∑ log(?𝑖)𝑛𝑖=1?(𝜃) =?𝜃 + 1− ∑ log(?𝑖)𝑛𝑖=1𝐿?? ?𝜃 + 1− ∑ log(?𝑖)𝑛𝑖=1= 0𝜃̂??𝐸=?log(?𝑖)𝑛𝑖=1− 1b) MME ?(?) = ∫???(?)??−∞= ∫ ?2?𝜃2??𝜃0=2𝜃2∫ ?2??𝜃0=2𝜃2[?33]0𝜃=2𝜃2𝜃33=2𝜃3
𝐿?? ?̅=2𝜃3𝜽̂??𝑬=??̅?MLE ℒ(𝜃) = ∏ ??(?𝑖)𝑛𝑖=1= ∏2?𝑖𝜃2𝑛𝑖=1= (2𝜃2)𝑛∏ ?𝑖𝑛𝑖=1ℓ(𝜃) = ???? (

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