Develop a Gibbs sampling method to simulate

Assignment Help Applied Statistics

Reference no: EM132407903

Problem 1 - Suppose that X ∼ Beta(2, 2) distribution, whose density function is f(x) = 6x(1 - x) for 0 < x < 1. Use a) a rejection sampling method and b) an importance sampling method to evaluate Var(X). Explain how you do it and write two separate pseudo-codes to implement the two methods.

Problem 2 - (a) Develop a Gibbs sampling method to simulate from a bivariate vector (X₁, X₂) from a bivariate normal distribution with zero mean and unit variance for the marginals, but a non-zero correlation of rho (r) between the two components. Explain how you do it and write a pseudo-code to implement it. (b) Suppose that you have N copies of the random sample (vector) simulated from your code. Explain how to use them to evaluate E{(X₁+X₂)²} and also the median of the distribution of (X₁+X₂)².

Problem 3 - Let (W₁(t), . . . , W_n(t)) be n independent Brownian motions. Define the new process

B_n(t) = √(W₁²(t) + . . . + W_n²(t)).

This process, called a Bessel process, describes the distance to the origin of an n-dimensional Brownian motion. We can show that B_n satisfies the equation

dB_n(t) = dZ(t) + (n-1)/(2B_n(t))dt

where Z(t) is a suitable Brownian motion. Assume that B_n(0) ≠ 0. Explain how to simulate a sample path of such a Bessel process and write a pseudo-code to implement it.

Problem 4 - (a) State the Bayes theorem.

(b) A disease has an overall base rate p(D) = .001. A test for the disease has sensitivity S (i.e, p(test positive |D) = S and specificity Q (i.e., p(test - |no D) = Q). If a patient tests positive what is the Bayesian posterior P(D|test +)?

(c) Numerical examples. Suppose S = Q = .99. What is the posterior P (D |test +)? Suppose S = .99, Q = .90. What is the posterior P(D|test +)?

Note - Need show steps for algorithm and few R code.

Reference no: EM132407903

Questions Cloud

State the Bayes theorem : State the Bayes theorem. Suppose S = Q = .99. What is the posterior P (D |test +)? Suppose S = .99, Q = .90. What is the posterior P(D|test +)

The challenges based on the organizational structure : Provide suggestions to overcome the challenges based on the organizational structure, communication methods, and training.

Describe the competitive strategies : Describe the competitive strategies used by the company's main competitors. Which of these strategies are the most effective?

Leaders and Operations Management Assessment : Management and Operations Assignment Help and Solution, Leaders and Operations Management Assessment Writing Service - Higher National Diploma in Business

Develop a Gibbs sampling method to simulate : Develop a Gibbs sampling method to simulate from a bivariate vector (X1, X2) from a bivariate normal distribution with zero mean

Social media and even in conversation : Social media and even in conversation, do you feel that noting where data and other vital information being shared came from could alleviate confusion,

Describe the lead user concept in comparison : Please describe the "Lead User concept" in comparison to classical market research methods. What are the advantages and disadvantages of the concept?

Ethical dilemma associated with the production : Briefly describe the company and its product and the ethical dilemma associated with the production and distribution of its products.

Standards in the host country are lower : Is a multi-national corporation morally obliged to follow the safety guidelines of its home country when the standards in the host country are lower? Explain.

User Account

All Pages