Reference no: EM13839577

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, . . . , 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 = X₁ + . . . + 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 X₁, . . . , 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 = X₁ + . . . + X_N.

(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.