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 104 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 X1, . . . , XN be a random sample from a Poisson distribution Poi(λ). Denote by SN the sum of the random sample, that is, SN = X1 + . . . + XN.

(a) Show that SN ~ Poi(Nλ).

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

4. Let X1, . . . , XN be a random sample from a Poisson distribution Poi(λ). Denote by SN the sum of the random sample, that is, SN = X1 + . . . + XN.

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

(b) For the two-sided hypothesis test H0: λ = λ0 vs. H1: λ ≠ λ0, determine a suitable test statistic T, the distribution of T, and the critical region for the test.

5. Repeatedly (106 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 H0: λ = λ0 is rejected in favour of the two-sided alternative H1 : λ ≠ λ0 at the α = 0.05 significance level, when λ0 = 2. Give a typical proportion and supply your code.

6. Repeatedly (106 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 H0 : λ = λ0 is rejected in favour of the two-sided alternative H1: λ ≠ λ0 at the α = 0.05 significance level, over the range λ0 ∈ [0, 4]. Give a typical plot of the proportion of rejections against λ0 and supply your code. For λ0 = 2, your plot is an estimate of the power of the test to correctly reject H0 in favour of the alternative.

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