# Advanced Probability and Statistics Questions

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Hi, please only bid / accept if you are fully able to answer these questions. I need a full explanation so I can fully study the concepts! Please let me know if you need more time or you’d like to be paid more. I’m cautious because I have gotten incorrect explanations before. Thank you so much (:For 14-16, only one of the options is right and the rest wrong or only one is wrong and the rest right. The answer would be the one that is different.For example, if (a) is the answer to a question, there are two possiblities:(a) is correct and (b), (c), (d), (e) are incorrectOR(a) is incorrect and (b), (c), (d), (e) are correct

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14. Suppose that under Pg2 the random variables X1, X2, … are i.i.d. real random variables with
X1 ~ N(u,02), where u R is known. Which of the following estimators is an unbiased estimator
for o2?
[1 Point]
(A) ô7 = 2=X?
(B) 2 = n-1 2-1 X?
(C) ô2 = 121-1(X; – Xn)?.
(D) ôz = +21=1(Xi – u)2
(E) ôz = 1 (2-1X;)? – nu?.
15. Let X1, …, Xn be i.i.d. real random variables with values in [3, 4]. The geometric mean Zn =
1
converges in probability to a real number a E R. What is its value? [1 Point]
(II>=1
(A) a = log (E[X]).
(B) a = E[X1]
(C) a=
E[log(X1)]
(D) a = exp (E[log(X1)]).
(E) a = exp (E[X2]).
16. A company sells bottled water. The maximal admissible amount of iron contained in such water
bottles is 1 mg. 25 bottles from the company are tested, and one finds for the amount of iron (in mg)
25
1
X 25 =
1
25
25
i=1
ΣΧ, = 1.1,
S25
(X; – X25)2 = 0.2.
25 – 1
i=1
We assume the amounts of iron in each bottle X1, …, X25 are realizations of i.i.d. real random
variables with X1 ~ N(0,0%), where o2 = 0.25 is known. We want to test
Ho:u=1,
against Hiu > 1
at a 2.5% level of significance. Let u1-a denote the (1 – a)-quantile of the standard normal
distribution and tk,1-e the (1 – a)-quantile of the t-distribution with k degrees of freedom. Which
of the following statements is true for the optimal test?
Hint: You may use the values U0.975 = 1.95996, 20.025 = -1.95996, t24,0.975 = 2.0639, t24,0.025 =
-2.0639.
[1 Point]
(A) The test rejects H, if X25 2 u + 25 +0.975, and H, is not rejected.
(B) The test rejects H, if X25 2H + Säg t24,0.975, and H, is not rejected.
(C) The test rejects H, if X25 2 4 + Szt24,0.025, and H, is rejected.
(D) The test rejects H, if X25 2H + U0.025, and H, is rejected.
(E) The test rejects H, if X25 2 + U0.975, and H, is rejected.
30. Let L1, …, Ln denote the life-times of n light bulbs. We assume L1, …, Ln to be i.i.d. with mean
u and standard deviation o(Li) = V Var[Lı] = 6 (in months). Use the central limit theorem to
determine approximately the probability that In = 121-1 Li deviates by less than a month from
M, i.e. P[|In – ul < 1], if n = 10. Hint: You can use freely available calculators for the values of 0(z) (the cumulative distribution function of N (0,1)) online, see for instance: https://homepage.divms.uiowa.edu/ mbognar/applets/normal.html Answer: (1 Point] [4 Points] Detailed explanation: Purchase answer to see full attachment Tags: Unbiased Estimator random variables estimators Real random variables maximum admissible amount User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

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