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Q1: 3. Joint probability distribution of several random variables: continuous case. Ex-amples of multivariate distributions.See Answer
Q2: 5. Probability distribution of a sum of two integer random variables. Examples foridentically distributed summands with Poisson, geometric, uniform distributions.See Answer
Q3: 7. Mathematical expectation (both discrete and continuous cases). Its properties(with proofs). Examples.See Answer
Q4: 9. Covariance and correlation (both discrete and continuous cases), their properties.Examples.See Answer
Q5: 10. Binomial distribution: its origins, mathematical expectation, variance.See Answer
Q6: 13. Hyper-geometric distribution: its origins, mathematical expectation, variance (withproofs).See Answer
Q7: 14. Geometric distribution: its origins, mathematical expectation, variance (with proofs).See Answer
Q8: 15. Continuous uniform distribution, its mathematical expectation, variance (withproofs).See Answer
Q9: 24. Simple linear regression (Gauss-Markov model): estimation, prediction.See Answer
Q10: 18. De Moivre – Laplace theorem. Central limit theorem. Examples (with proofs).See Answer
Q11:2. Suppose we roll two six sided die independently: Dicel: Fair dice. Dice2: Unfair, with probability 0.2 for landing 1, 0.4 for landing 5, and 0.1 for the rest of the 4 numbers. (a) What is the probability that at least one is a 5. (b) What is the probability that at least one is a 5 and that the two numbers that show are different. (c) What is the probability that at least one is a 5 and the sum of the two numbers is strictly greater 8. Show all work.See Answer
Q12:4. Suppose we are working with a circular dart board with 6 concentric regions labelled 1 (outermost) to 6 (disc at the center). Assume that the dart board has radius r and that a dart thrown at the board always hits the board. It is easy to see that the probability of scoring i points on a single throw is (a) Show that P(scoring i points) = Area of region i Area of the dart board P(scoring i points) = (7-i)²-(6-1)² 6² Note that this probability is independent of the radius of the dartboard, why? (b) Show that function constructed above satisfies the probability axioms. (c) Show that the probability function constructed above is an increasing function of i. 5. Suppose P is a probability function on the pair (sample space, sigma algebra): (S,B). Using only the three defining properties of P, and for any A, B € B, show that: (a) P(A) = 1- P(A). (b) P(AUB) = P(A) + P(B) - P(ANB). (c) If A C B, then P(A) ≤ P(B).See Answer
Q13:FORUM DESCRIPTION Part (a): Create a discrete probability distribution using the generated data from the following simulator: Anderson, D. Bag of M&M simulator. New Jersey Factory. Click on the simulator to scramble the colors of the M&Ms. Next, add the image of your generated results to the following MS Word document: Discrete Probability Distributions.docx. Use the table in this document to record the frequency of each color. Then, compute the relative frequency for each color and include the results in the table. Part (b): Compute the mean and standard deviation (using StatCrunch or the formulas) for the discrete random variable given in the table from part (a). Include your results in the MS Word document. Part (c): Add a screenshot of the entire completed MS Word document to the discussion board as part of your discussion response (do NOT upload the MS Word document). Part (d): Comment on your findings in your initial response and respond to at least 2 of your classmate's findings. Grading InformationSee Answer
Q14:Question 1 Advertising researchers have developed a theory that states that commercials that appear in violent television shows are less likely to be remembered and will thus be less effective. After examining samples of viewers who watch violent and nonviolent programs and asking them a series of five questions about the commercials, the researchers produced the following probability distributions of the number of correct answers. Viewers of violent shows 0 0.36 X P(x) Viewers of nonviolent shows X P(x) 1 0.22 0 0.15 1 0.18 2 0.20 2 0,23 3 0.09 3 0,26 4 0.08 4 0.10 5 0.05 5 0.08 a) (10 points) Calculate the mean and standard deviation of the number of correct answers among viewers of violent television programs. b) (10 points) Calculate the mean and standard deviation of the number of correct answers among viewers of nonviolent television programs.See Answer
Q15:Question 2 In a typical month, an insurance agent presents life insurance plans to 40 potential customers. Historically, one in four such customers chooses to buy life insurance from this agent. Based on the relevant Binomial distribution, answer the following questions about X, the number of customers who will buy life insurance from this agent in the coming month: a) (5 points) What is the probability that X is exactly 5? 2 OPRE 3360- Managerial Methods in Decision Making Under Uncertainty b) (5 points) What is the probability that X is no more than 10? c) (5 points) What is the probability that X is at least 20? d) (5 points) Determine the mean and standard deviation of X. Homework 2See Answer
Q16:Question 3 Hits on a personal website occur quite infrequently. They occur randomly and independently with an average of five per week. a) (8 points) Find the probability that the site gets 10 or more hits in a week. b) (12 points) Determine the probability that the site gets 20 or more hits in 2 weeks.See Answer
Q17:Question 4 Let the random variable Z follow a standard normal distribution. Find the following probabilities. a) (4 points) P(Z<1.72), b) (4 points) P(Z>1.1), c) (4 points) P(1.1<Z<1.72), d) (4 points) P(Z<-1.1), e) (4 points) P(-0.23<Z$2.00).See Answer
Q18:Question 5 (20 points) Assume that the weight of a typical male follows normal distribution with 180 lb and a-30 lb. What fraction of males weigh more than 200 pounds?See Answer
Q19:1. UNDERWATER MORTGAGE Twenty percent of US mortgages are "underwater" (The Boston Globe, March 5, 2009). A mortgage is considered underwater if the value of the home is less than what is owed on the mortgage. Suppose 100 mortgage holders are randomly selected. a) What is the probability that more than 20 of the mortgages are underwater? b) What is the probability that at least 25 of the mortgages are underwater?See Answer
Q20:2. CHEATING ON AN EXAM A professor has learned that three students in her class of 20 will cheat on the exam. She decides to focus her attention on four randomly chosen students during the exam. a) Find the probability that she finds at least one of the students cheating if she focuses on four randomly chosen students? b) Find the probability that she finds at least one of the students cheating if she focuses on six randomly chosen students?See Answer

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