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What is the approximate distribution of sn for large n -

**Question.**

Complete assignment with solutions on words or pdf. graph and tables should be included if needed. codes for mathlab should be included as well. All solutions should be presented on mathlab

1. A certain multiple choice exams has 30 questions, each providing 3 choices. To pass the exam one needs at least 20 out of 30 correct answers.

(a) Suppose a student knows the answers to 27 questions for certain, and fills in the remaining three questions \at random". What is the probability that the student will get full marks?

(b) Another student knows only 10 questions for certain and, like the student in (a), fills in the remaining questions at random. Use the Central Limit Theorem to approximate the probability that this student will pass.

2. Simulate 10^{4 }outcomes (X, Y), where

X = Rcos(Θ) , Y = Rsin(Θ),

with Θ ~ U(A) where A = [0, Π/4) ∪ [ Π/2, 3Π/4) ∪ [Π, 5Π/4) [ [3Π/2, 2Π), and R = 5 + S with S ~ Exp(1) independently of Θ. Provide a scatter plot of typical output, and give the corresponding calculated (empirical) correlation.

Comment on the correlation and dependence of (X, Y ). Supply your code.

3. Let X_{1}, . . . , X_{N} be a random sample from a Poisson distribution Poi(λ). Denote by S_{N} the sum of the random sample, that is, S_{N} = X_{1} + . . . + X_{N}.

(a) Show that S_{N} ~ Poi(N_{λ}).

(b) Using the central limit theorem, what is the approximate distribution of S_{N} for large N?

4. Let X_{1}, . . . , X_{N} be a random sample from a Poisson distribution Poi(λ). Denote by S_{N} the sum of the random sample, that is, S_{N} = X_{1} + . . . + X_{N}.

(a) Using Q3(b), determine an approximate 1 - α stochastic confidence interval for λ.

(b) For the two-sided hypothesis test H_{0}: λ = λ0 vs. H_{1}: λ ≠ λ0, determine a suitable test statistic T, the distribution of T, and the critical region for the test.

5. Repeatedly (10^{6} times) simulate a random sample of size N = 20 from Poi(2) and record the sum. Using Q4(b), compute the proportion of times that the null hypothesis H_{0}: λ = λ_{0} is rejected in favour of the two-sided alternative H_{1} : λ ≠ λ_{0 }at the α = 0.05 significance level, when λ_{0} = 2. Give a typical proportion and supply your code.

6. Repeatedly (10^{6} times) simulate a random sample of size N = 20 from Poi(2) and record the sum. Using Q4(b), compute the proportion of times that the null hypothesis H_{0} : λ = λ_{0} is rejected in favour of the two-sided alternative H_{1}: λ ≠ λ_{0 }at the α = 0.05 significance level, over the range λ_{0} ∈ [0, 4]. Give a typical plot of the proportion of rejections against λ_{0} and supply your code. For λ_{0} = 2, your plot is an estimate of the power of the test to correctly reject H_{0} in favour of the alternative.

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